MAPLE commands
for Differential Equations with Linear Algebra
MAT2705created 2001 by bob jantzen; version15-apr-2018 [Villanova University]
<execute worksheet with toolbar icon !!! to restore output>This worksheet summarizes the commands that are helpful in using MAPLE to as a supporting tool in this course. In a perfect world, students could refer to examples here when necessary to do assigned homework problems, but the reality is that no one has time to look at this worksheet. Fortunately the right click context sensitive menus plus palette insertion of Clickable Calculus enables elementary users to not have to deal very much with Maple commands. Although many of these commands may not be useful in this course, it is useful for instructors to at least be aware of them in case some might prove useful to the activities they choose to do in guiding student learning.
Any input line can be changed from
Maple text (1d math input), as in: > f := x -> x^2;to Maple math (2d math input) , as in: > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUklbXN1cEdGJDYlRk8tRiM2JS1JI21uR0YkNiRRIjJGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOQ==
or back by using the "Format Menu, Convert To" option. Many of the results below may be obtained with palette entry and right click menu selection. You may also enter the 1d input lines typing into 2d input, using the 2d input features to build up math expressions which occur in the command inputs. You can also click on the leftmost lower toolbar line icon "Text" while the cursor is in an input region, and force it to accept 1d input commands as they appear in a book.
Pressing "Shift, Enter" simultaneously allows you to put in a new line of Maple input, and semi-colons allow you to separate Maple inputs in 2d math, while the colon will suppress the output, which is usually what you want to do when loading a package to avoid the long list of commands which are loaded into memory. A collection of useful tips for using Standard Maple and 2d math can be found at the webpage:
http://www.villanova.edu/artsci/mathematics/resources/maple/using/mapletips.htmFor example, occasionally an underscore is useful in naming quantities, but in 2d math, the underscore converts to subscript mode, so first typing backslash " \134 " disables this feature and allows an underscore character to be entered immediately after the backslash.This worksheet can be executed as a whole using the !!! icon on the toolbar, but you must close any popup windows that appear (click on the button "Close" or "Cancel"), and select "function definition" in the first autocomplete prompt query by clicking on it. Then you can scroll back to the top and read it. Alternatively one can execute each input one at a time working down through the worksheet, or load the packages and go directly to the section that interests you.
WARNING: Not everything in this worksheet is useful for a short course on differential equations with linear algebra, but at least it shows the reader what is available on the topic.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=loading Maple packagesFor special differential equations and linear algebra commands we must load the packages:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSV3aXRoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUShERXRvb2xzRidGNEY3Rj5GPkY+LUY7Ni1RIjpGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGam5GPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSV3aXRoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUS5MaW5lYXJBbGdlYnJhRidGNEY3Rj5GPkY+LUY7Ni1RIjpGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGam5GPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSV3aXRoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUShTdHVkZW50RidGNEY3LUZVNiYtRiM2JC1GLDYlUS5MaW5lYXJBbGdlYnJhRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YrRj5GPkY+LUY7Ni1RIjpGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGZ29GPg==The Student[LinearAlgebra] package, which has the additional Tutor commands, is slightly more pedagogical than the LinearAlgebra package, and for our purposes it suffices.
[If we want the Column or Row commands to extract a given row column or row from a matrix, we need to load the LinearAlgebra package.]
Note that putting a colon after a load package input supresses the list of commands loaded into memory.JSFHcalculusMAPLE expressions and functionsMaple functions (preferable to work with a function and be able to use function notation, but sometimes unapply is necessary in place of the arrow function definition), whose values are MAPLE expressions:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEieEYnRi9GMi1GNjYtUS0mUmlnaHRBcnJvdztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVi1JJW1zdXBHRiQ2JUZPLUkjbW5HRiQ2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRidGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zhby8lJmRlcHRoR0Znby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUSJGRidGL0YyRjUtRiw2JVEodW5hcHBseUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnRlgtRjY2LVEiLEYnRjlGO0Zfb0ZARkJGREZGRkhGYG8vRk5RLDAuMzMzMzMzM2VtRidGT0Y5RjlGXG9GYm9GKy1GZnA2JC1GIzYkRk9GOUY5RlxvRmJvRl9wRmJxRmpwRjk=MAPLE expressions do not work with function notation: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 in this next example which is like MathCad input, by responding to the ambiguity prompt popup box asking it to interpret this input as a function definition, does work: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJHRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg==Maple arrow functions can be inserted directly from the Expression palette, and stringnames can be assigned with the := assignment statement to any Maple input lines to use them later on. % and %% stand for the last and next to last Maple outputs, which can be useful in feeding one output into another in a single execution group (all the inputs within one left square bracket on the left margin, all input commands within such a group are executed together).differentiation and integrationfirst derivativesMAPLE function and D operator: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNipGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Y4LyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUY0NiZGNkY5RjwvRj9RJWF1dG9GJy1GLDYlUSJERicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRImZGJy9GSVEldHJ1ZUYnL0ZMUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRkpGS0ZLLUZPNiQtRiM2JS1JI21uR0YkNiRRIjJGJ0ZLRlpGS0ZLRlpGS0YrRlpGS0YrRlpGSw==MAPLE expression and diff operator (first start with the "evaluate f at" icon, then from the palette insert the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for the standin symbol LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, ..., nahh! too much effort):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evaluating a MAPLE function leads to an expression: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palette entry in 2d input uses 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 still need to think about parentheses (no implied parentheses in the previous expression)!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 can still use functions in 2d input and even the prime notation: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 derivativesMAPLE function and D operator: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 expression and diff operator (you can also use palettes but it is somewhat cumbersome):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evaluating a MAPLE function leads to an expression: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With palette input, you simply repeat the derivative but the prime function notation is much more efficient up to 3 primes (it is a bit hard to count more than 3 prime superscripts even though the notation now works), after which you need to use the standard parentheses-surrounded integer superscripts):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with palette 2d input (you must put in the constant of integration by 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on the gridlines by clicking on the plot and then on the context activated toolbar line icon for a black grid, or using the explicit option "gridlines=true" can often be helpful in making a graph more useful.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzY1LUY2NiYtRiM2KC1JI21vR0YkNi1RJHNpbkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUY2NiQtRiw2JVEieEYnRi9GMkZCLUY/Ni1RIixGJ0ZCL0ZFUSZmYWxzZUYnL0ZIRjEvRkpGW28vRkxGW28vRk5GW28vRlBGW28vRlJGW29GUy9GV1EsMC4zMzMzMzMzZW1GJy1GIzYoLUYsNiNRIUYnLUY/Ni1RJGNvc0YnRkJGREZHRklGS0ZNRk9GUUZTRlYtRj82LVEwJkFwcGx5RnVuY3Rpb247RidGQkZqbi9GSEZbb0Zdb0Zeb0Zfb0Zgb0Zhb0ZTRlZGWEZmb0ZCRmZvRkJGQi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZnbkZaLUY/Ni1RIj1GJ0ZCRmpuRl9wRl1vRl5vRl9vRmBvRmFvL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRmpwLUkjbW5HRiQ2JFEiMEYnRkItRj82LVEjLi5GJ0ZCRmpuRl9wRl1vRl5vRl9vRmBvRmFvL0ZUUSwwLjIyMjIyMjJlbUYnRlYtRl1xNiRRIjJGJ0ZCLUY/Ni5RMSZJbnZpc2libGVUaW1lcztGJy8lK2V4ZWN1dGFibGVHRltvRkJGam5GX3BGXW9GXm9GX29GYG9GYW9GU0ZWLUYsNiVRJSZwaTtGJy9GMEZbb0ZCRmduRmhxLUYsNiVRJmNvbG9yRidGL0YyRmZwLUY2NiYtRiM2Ji1GLDYlUSRyZWRGJ0YvRjJGZ24tRiw2JVElYmx1ZUYnRi9GMkZCRkJGYHBGY3BGZ24tRiw2JVEqZ3JpZGxpbmVzRidGL0YyRmZwLUYsNiVGMUYvRjJGQkZCRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1GIzYmLUYsNiVRJXBsb3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GPFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1JKG1mZW5jZWRHRiQ2JC1GIzYzLUY/Ni1RIltGJ0ZCL0ZFRjpGRy9GSkY6RktGTUZPRlEvRlRRLDAuMTY2NjY2N2VtRicvRldGXW9GKy1GIzYqRistRj82LVEkc2luRidGQi9GRVEmdW5zZXRGJy9GSEZlby9GSkZlby9GTEZlby9GTkZlby9GUEZlby9GUkZlb0ZTRlZGPi1GWTYkLUYsNiVRInhGJ0Y4RjtGQi1GPzYtUSIsRidGQkZEL0ZIRjpGSUZLRk1GT0ZRRlMvRldRLDAuMzMzMzMzM2VtRictRiM2KEYrLUY/Ni1RJGNvc0YnRkJGZG9GZm9GZ29GaG9GaW9Gam9GW3BGU0ZWRj5GXHBGK0ZCRitGQi1GPzYtUSJdRidGQkZqbkZHRltvRktGTUZPRlFGXG8vRldRLDAuMTExMTExMWVtRidGYXBGXnAtRj82LVEiPUYnRkJGREZHRklGS0ZNRk9GUS9GVFEsMC4yNzc3Nzc4ZW1GJy9GV0ZlcS1JI21uR0YkNiRRIjBGJ0ZCLUY/Ni1RIy4uRidGQkZERkdGSUZLRk1GT0ZRL0ZUUSwwLjIyMjIyMjJlbUYnRlYtRmhxNiRRIjJGJ0ZCLUY/Ni5RMSZJbnZpc2libGVUaW1lcztGJy8lK2V4ZWN1dGFibGVHRkZGQkZERkdGSUZLRk1GT0ZRRlNGVi1GLDYlUSUmcGk7RicvRjlGRkZCRmFwLUYsNiVRJmNvbG9yRidGOEY7RmFxLUYsNiVRJmJsYWNrRidGOEY7RkJGQkZCRitGQkYrRkJGK0ZCYou can also plot by right clicking on the output of a single function once you input it and execute it, or on a sequence of function expressions separated by commas. Try it, then right click on the graph to choose "Axes, Properties" to change the viewing window etc:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR0Y7L0YwUSdpdGFsaWNGJ0YyLUklbXN1cEdGJDYlRkwtRiw2JFEiMkYnRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZNNiNRIUYnRi8=solving equationsYou can enter an expression and right click "solve" to find its solutions, if it can be solved exactly: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Otherwise one must solve numerically only after plotting the function and choosing an interval where exactly one solution exists, unless it is a polynomial, when all solutions will be found.
Here is a blind right click numerical solution, but if the equation is not a polynomial equation, it gives up when it finds one solution, which may not be the one you want: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1st order deqsdsolve [diff or D notation]With the MAPLE expression derivative diff notation, general solution and initial value problem solution: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Here is a cute optional command for checking MAPLE which substitutes the solution into the LHS - RHS of the differential equation so if the result is 0, the expression is a solution: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With the MAPLE function derivative D notation: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2d input shortcuts (prime notation for derivatives)In 2d input mode, prime superscripts are interpreted as derivatives and right-clicking on the output allows you to "Solve DE"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Apparently if you do not specify the independent variable, it assumes it to be x but the View menu, Typesetting Rules window explains how to change this: see bottom left "Differential options", "prime derivatives".If you try to change it, it switches to D notation. These interface developers are a bit sloppy, eh? Or perhaps they did not intend this to work this way.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This line will set the prime derivative variable to t without passing through the Typesetting Rule window (this is too complicated for students, not even I use it anymore):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However, by just using function notation like this, you don't need to reset the default, althought the output uses the D derivative notation instead of the d/d notation.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We restore the default prime derivative variable: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 DEplot [directionfield, numerical solution graphs]All of these 1d inputs can be entered directly in 2d input mode as well:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSV3aXRoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUShERXRvb2xzRidGNEY3LyUrZXhlY3V0YWJsZUdGQkY+Rj5GZm5GPi1GOzYtUSI6RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRlxvRmZuRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVElZGVxMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJWRpZmZGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSJ5RidGL0YyLUZTNiQtRiM2JS1GLDYlUSJ4RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJ0ZobkZbb0Y5RjktRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpby1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGOS1GIzYlRmhuRi9GMi8lLmxpbmV0aGlja25lc3NHRmFwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhwLyUpYmV2ZWxsZWRHRj0tRjY2LlExJkludmlzaWJsZVRpbWVzO0YnRltvRjlGO0Y+RkBGQkZERkZGSEZhby9GTkZib0ZXRlotRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GXnBGW29GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEnREVwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2LC1GLDYlUSVkZXExRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GIzYnLUYsNiVRInlGJ0Y2RjlGPC1GVzYkLUYjNiUtRiw2JVEieEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmpvRkBGaG4tRiM2KEZnby1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmJwLUYjNihGKy1GIzYmLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZccS1JI21uR0YkNiRRIjRGJ0ZARmpvRkAtRj02LVEjLi5GJ0ZARkJGRUZHRklGS0ZNRk9GW3FGVEZecUZqb0ZARitGam9GQEZobi1GIzYoRmBvRl5wRmRwRitGam9GQEYrRmpvRkBGQEZqb0ZARitGam9GQEYrRmpvRkA=Looking at the help for DEplot shows this option for the number of grid points in each direction: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This has trouble with division by 0 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= for the numerical solution curves so we only show a region away from there; we need to choose an initial data point for each solution curve:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEmaW5pdHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictSShtZmVuY2VkR0YkNiYtRiM2Jy1JI21uR0YkNiRRJDAuNUYnRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVVEsMC4zMzMzMzMzZW1GJy1GaG42JFEiMEYnRkAvJStleGVjdXRhYmxlR0ZERkBGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zbby1GWTYmLUYjNictRmhuNiRRIjFGJ0ZARltvLUZobjYkUSIyRidGQEZmb0ZARkBGaG9GW3BGZm9GQEYrRmZvRkBGK0Zmb0ZALUY9Ni1RIjtGJ0ZARkJGXm9GR0ZJRktGTUZPRl9vRlRGZm9GQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjYvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjI2JkY0RjdGOi9GPVElYXV0b0YnLUYsNiVRJ0RFcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzY4LUYsNiVRJWRlcTFGJ0ZGRkktSSNtb0dGJDYtUSIsRicvRkpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRmZuLyUqc3ltbWV0cmljR0Zmbi8lKGxhcmdlb3BHRmZuLyUubW92YWJsZWxpbWl0c0dGZm4vJSdhY2NlbnRHRmZuLyUnbHNwYWNlR0Y5LyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGRkZJLUZNNiQtRiM2JS1GLDYlUSJ4RidGRkZJLyUrZXhlY3V0YWJsZUdGZm5GWEZYRlRGX3AtRlU2LVEiPUYnRlhGWi9GaG5GZm5GaW5GW29GXW9GX29GYW8vRmRvUSwwLjI3Nzc3NzhlbUYnL0Zmb0ZpcC1JI21uR0YkNiRRJDAuMUYnRlgtRlU2LVEjLi5GJ0ZYRlpGZ3BGaW5GW29GXW9GX29GYW8vRmRvUSwwLjIyMjIyMjJlbUYnL0Zmb0Y5LUZccTYkUSI0RidGWEZURmhvRmRwLUZVNi1RKiZ1bWludXMwO0YnRlhGWkZncEZpbkZbb0Zdb0Zfb0Zhb0ZicS9GZm9GY3FGZXFGX3FGZXFGVC1GTTYmLUYjNiUtRiw2JVEmaW5pdHNGJ0ZGRklGYnBGWEZYLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrRmJwRlhGWEZicEZYRitGYnBGWA==For some reason, you cannot add the gridlines=true option here but you can get it from the plot context toolbar line icon, this can also be very useful.
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following only plots the numerical solution curves (the numpoints option makes it less jagged near the vertical axis):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 [displaying solutions]We need the plots package for "display"ing plots together.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with the directionfield DEplot: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and compare with the numerical solution ( you can right-click on a yellow solution curve and change its color to blue or whatever to make it more well 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order linear deqsdsolve [diff or D notation]With the MAPLE expression derivative diff notation, general solution and initial value problem solution: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Here is a cute optional command for checking MAPLE: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With the MAPLE function derivative D notation: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2d input shortcuts (prime notation for derivatives)In 2d math, prime superscripts are interpreted as derivatives and right-clicking on the output allows you to "Solve 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line will set the prime derivative variable to t without passing through the Typesetting Rule 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can also directly use the command as long as we use set notation for the the set of equations to be 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restore the default prime derivative 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DEplot [numerical solution 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is a bit jagged (look at the extrema) so an option smooths this out: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dsolve [displaying solutions]We can just list the right hand sides of the dsolve solutions in a plot command: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 numbers, exponentials, deqsFor constant coefficient 2nd or higher order differential equations, one needs to use complex exponentialscomplex arithmeticMaple has some obvious commands for dealing with complex number algebra. One can either type in "Re" or "Im" or pick their symbols \342\204\234 and \342\204\221 from the common symbols 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 Maple does not know if the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is real or not, the Real and Imaginary part commands do not simplify in the next complex expression, but the command evalc assumes that the variables in an expression are real and conveniently evaluates the expression to its real and imaginary parts as if they were: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSZldmFsY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkZGJ0Y0RjcvRjhRJ25vcm1hbEYnRkJGQkYrRitGK0YrRkI=Maple automatically simplifies complex numbers to their real plus imaginary part 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But Maple stops this simplification when unknown variable names appear, since they could be complex and the result would change: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while evalc will assume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is a real variable and go ahead and do it:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmZXZhbGNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJ0Y9Rj0=JSFHfactoring, solving polynomial equations with complex rootsBecause we like real expressions, factor does not factor quadratics with complex roots, but solve finds all the roots: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For this fourth order characteristic equation (its roots are the four fourth roots of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiNEYnRjVGNUYrRjU=):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlhGOi1GRDYtUSI9RidGPkZHRkpGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRmluLUY7NiRGQkY+LUZENi1RIjtGJ0Y+RkcvRktGNEZMRk5GUEZSRlQvRldRJjAuMGVtRidGam4tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGYm8vJSZkZXB0aEdGaG8vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRi82JlEmc29sdmVGJ0YyLyUrZXhlY3V0YWJsZUdGSUY1LUkobWZlbmNlZEdGJDYlLUYjNiotRiw2JS1GLzYmRjFGMkZjcEY1LUYjNiQtRjs2JUY9RmNwRj5GPkZALUZENi5GRkZjcEY+RkdGSkZMRk5GUEZSRlRGVkZZRmBxLUZENi5GZ25GY3BGPkZHRkpGTEZORlBGUkZURmhuRmpuLUY7NiVGQkZjcEY+LUZENi5RIixGJ0ZjcEY+RkdGYG9GTEZORlBGUkZURmFvL0ZaUSwwLjMzMzMzMzNlbUYnRlxxRj5GY3BGPkY+[This next input is a Maple trick, don't try this yourself---the map command is worth getting to know for serious Maple users, which by definition MAT2705 students are not:]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The real and imaginary parts of the corresponding complex exponentials give the real solutions of the related constant coefficient homogeneous differential equation: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JSFHmatrices (LinearAlgebra) Maple handles vectors and matrices using the package LinearAlgebra, which is associated with the Matrix palette insertion 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 the version for students which is more pedagogical: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 period "." can be used for matrix multiplication even without loading either package, and even better, a simple space between two matrices should result in their matrix product (only in mysterious circumstances does the space fail and the period force the issue).JSFHhelp pages and examplesBy deleting the number sign "#" from the first character position in the following 4 input lines, one calls up the help pages for these packages. The number sign "#" turns off Maple from acting on the rest of the input line so that when you execute the whole worksheet, a dozen or more separate help documents do not get generated from these various calls to help pages in this worksheet.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEvP0xpbmVhckFsZ2VicmFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEwP2V4YW1wbGVzL2luZGV4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=Open the example section on Algebra. One good example which describes the syntax for easy entry of rows and columns in matrices is:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFDP2V4YW1wbGVzL0xpbmVhckFsZ2VicmFDb21wdXRhdGlvbkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yvlisting the LinearAlgebra commandsWe only need a few of these commands in our course. It is better to suppress this list with the colon when we load it (sometimes I load LinearAlgebra to extract columns from a matrix with "Column", not available in the Student version)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1GLDYjUSFGJ0Y9RkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKFN0dWRlbnRGJ0YvRjItRjY2Ji1GIzYkLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvRjNRJ25vcm1hbEYnRkQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGREZELUYsNiNRIUYnRkQ=Wow, way too many, eh? Fortunately we only need a few of these commands.entering MatricesYou have 4 choices to enter a matrix (the LinearSolveTutor only allows you to edit up to 5 rows or columns):
1) with the Matrix palette,
2) with the Matrix command,
3) or with the shortcut < | ,> notation, with " | " as a horizontal separator, and " , " as a vertical separator, either by rows or columns: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entered as 3 rows of 4 entries each: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 as 3 rows of 4 entries each: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entered as 4 columns of 3 entries each: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 access a matrix element, you use
"row,column" subscript notation (achieved by using eh underscore character to shift down to subscript level and the right arrow to climb back up afterwards if necessary) or
"[row,column]" square bracket notation:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEiM0YnL0Y5USdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0ZBLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY+NiRRIjRGJ0ZBRkEvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRkE=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 can also enter random matrices of any size with the Matrix palette or the commands RandomMatrix or RandomVector, giving its size:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS1SYW5kb21NYXRyaXhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiM0YnRj4tRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnRllGPkY+Rj5GK0Y+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS1SYW5kb21WZWN0b3JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiM0YnRj5GPkY+Rj5GK0Y+With your cursor in a matrix in the input region, you can do "control shift C"to add another column to the right of the cursor (good for adding a zero column to an existing row reduced coefficent matrix to use BackwardSubstitute to solve the system) or "control shft R" to add another row below the cursor.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scalar multiplication, matrix addition, linear 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matrix 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJCRidGL0YyRjk=vectors and matrices in MAPLE, matrix 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Matrix multiplication:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJYRidGL0YyRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNictSSNtbkdGJDYkUSIxRidGOS1GNjYtUSJ8Z3JGJ0Y5RjtGPi9GQUYxRkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZ24tRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl1vLUZVNiRRIjJGJ0Y5RjlGOS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJBRidGL0YyRjk=We can convert back and forth between vectors (shown as columns) and row matrices (shown as rows):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We can also build up larger matrices from smaller matrices, if the dimensions are consistent.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtbkdGJDYkUSIxRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUZVNiRRIjJGJ0Y5RlgtRlU2JFEiM0YnRjlGOUY5LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GNjYtUSI7RidGOUY7RmVuRkBGQkZERkZGSEZmbkZNLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGZm4vRk5GZ24tRiw2JVEiVkYnRi9GMkY1LUZQNiYtRiM2KC1GVTYkUSI0RidGOUZYLUZVNiRRIjVGJ0Y5RlgtRlU2JFEiNkYnRjlGOUY5RmBvRmNvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNiYtRiw2JVEiVUYnRi9GMi1GNjYtUSJ8Z3JGJ0Y5RjtGPi9GQUYxRkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZm4tRiw2JVEiVkYnRi9GMkY5RjkvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRjk=An alternative way of entering vectors uses Vector: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 are treated like column matrices.Sometimes a space between matrices will generate some rtable error that I have never understood, in which case a period in place of the space between matrices will always multiply them.transpose (switching rows and columns)The transpose interchanging rows and columns can be done with the command or the superscript short form: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We can enter vectors into a matrix as columns (natural) or rows (with the transpose)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRIlVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjJVRGJ0Y3RjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRisvRjtRJ25vcm1hbEYnRitGQw==special matrices 0 and 1The zero matrix or unit matrix of a given dimension are: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS9JZGVudGl0eU1hdHJpeEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIzRidGPkY+Rj5GPkYrRj4=solving linear equationsthe matrix reduction tutorsOnce you invoke the LinearSolveTutor, it asks you if you want to do Gauss Jordan or only Gaussian elimination (Gauss Jordan = RREF is preferred for us). We could have also started with either of these two tutors directly from the Tools, Tutor, LinearAlgebra, Solving Linear Systems Menu choice. You can copy any part of the tutor output window with mouse selection and control C, and then you can paste that back into the worksheet in a text region, saying to convert the MathML to 2d input when it prompts you.
If you started with an augmented matrix for a linear system, after reducing the matrix, you can choose the Solve the Equations, and then either Equations or Solution to then copy and paste the new right output window contents into a text region. See both below.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVExTGluZWFyU29sdmVUdXRvckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkFGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2I1EhRidGPQ==transferring from the tutor windows back into the worksheetIf you want to record both the reduction steps and the backsub solution from the tutor windows, you can open a second copy of Maple and mouse select Control C copy window contents in the Tutor one at a time to paste into a text region in the second worksheet and then copy and paste them all back into the original worksheet in a text region. Here I added some row operation notes to the reduction steps after each arrow, and copied both the equations and the solution as well.
[Unfortunately if you execute the whole worksheet, this section of paste math text will try to execute and generate an error. Ignore it.]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Equations: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: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Note that if you let Maple do all the steps, it will start by dividing by the upper left entry (it has no fear of fractions!)First the Equations form output, then the Solutions form output:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNictSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYoLUkkbXRkR0YkNigtRiM2J0YrLUYjNiZGKy1JI21uR0YkNiVRIjFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0ZDRkZGK0ZDRkYvJSlyb3dhbGlnbkdGLi8lLGNvbHVtbmFsaWduR0YuLyUrZ3JvdXBhbGlnbkdGLi8lKHJvd3NwYW5HRkIvJStjb2x1bW5zcGFuR0ZCLUY5NigtRiM2J0YrLUYjNiZGKy1GQDYlUSIwRidGQ0ZGRkNGRkYrRkNGRkZJRktGTUZPRlEtRjk2KC1GIzYnRistRiM2J0YrLUkmbWZyYWNHRiQ2KC1GQDYlUSQzNjBGJ0ZDRkYtRkA2JVEiN0YnRkNGRi8lLmxpbmV0aGlja25lc3NHRkIvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaW8vJSliZXZlbGxlZEdGRUYrRkNGRkYrRkNGRkZJRktGTUZPRlFGSUZLRk0tRjY2KEZTRjgtRjk2KC1GIzYnRistRiM2J0YrLUZdbzYoLUZANiVRJS0yNDNGJ0ZDRkYtRkA2JVEjMTRGJ0ZDRkZGZW9GZ29Gam9GXHBGK0ZDRkZGK0ZDRkZGSUZLRk1GT0ZRRklGS0ZNLyUmYWxpZ25HUSVheGlzRicvRkpRKWJhc2VsaW5lRicvRkxGaW8vRk5RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR1EldHJ1ZUYnLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGW3IvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZnIvJSZmcmFtZUdGZnIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZFLyUtZXF1YWxjb2x1bW5zR0ZFLyUtZGlzcGxheXN0eWxlR0ZFLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmNyRkNGRi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1JI21vR0YkNi5RLSZSaWdodEFycm93O0YnRkNGRi8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGZXQvJSlzdHJldGNoeUdGaHEvJSpzeW1tZXRyaWNHRmV0LyUobGFyZ2VvcEdGZXQvJS5tb3ZhYmxlbGltaXRzR0ZldC8lJ2FjY2VudEdGZXQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZkdS1GMDYlLUYzNjYtRjY2Ji1GOTYoLUYjNidGKy1GIzYnRistRiM2KUYrLUklbXN1YkdGJDYlLUYsNiZRInhGJy8lJ2l0YWxpY0dGaHFGQy9GR1EnaXRhbGljRidGPy8lL3N1YnNjcmlwdHNoaWZ0R0Zlbi1GYHQ2LlEpJmVxdWFscztGJ0ZDRkZGY3RGZnQvRml0RmV0Rmp0Rlx1Rl51RmB1RmJ1RmV1RlxvRitGQ0ZGRitGQ0ZGRitGQ0ZGRklGS0ZNRk9GUUZJRktGTS1GNjYmLUY5NigtRiM2J0YrLUYjNidGKy1GIzYpRistRmZ2NiVGaHYtRkA2JVEiMkYnRkNGRkZfd0Zhd0ZmcEYrRkNGRkYrRkNGRkYrRkNGRkZJRktGTUZPRlFGSUZLRk1GXnFGYXFGY3FGZHFGZnFGaXFGXHJGXnJGYXJGZHJGZ3JGaXJGW3NGXnNGYHNGYnNGZHNGZ3NGQ0ZGRitGQ0ZGLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2I1EhRictSSZtdGV4dEdGJDYlUSNJZkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYxLjBlbUYnLyUmZGVwdGhHRj4vJSpsaW5lYnJlYWtHUShub2JyZWFrRidGKy1JKG1mZW5jZWRHRiQ2Jy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1GIzYnRistRiM2JkYrLUkjbW5HRiQ2JVEkLTE2RidGM0Y2RjNGNkYrRjNGNi8lKXJvd2FsaWduR0YuLyUsY29sdW1uYWxpZ25HRi4vJStncm91cGFsaWduR0YuLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRl1vLUZRNigtRiM2J0YrLUYjNiZGKy1GWDYlUSQtNTBGJ0YzRjZGM0Y2RitGM0Y2RmVuRmduRmluRltvRl5vRmVuRmduRmluLUZONictRlE2KC1GIzYnRistRiM2JkYrLUZYNiVRIy05RidGM0Y2RjNGNkYrRjNGNkZlbkZnbkZpbkZbb0Zeby1GUTYoLUYjNidGKy1GIzYmRistRlg2JVEkLTIyRidGM0Y2RjNGNkYrRjNGNkZlbkZnbkZpbkZbb0Zeb0ZlbkZnbkZpbi8lJmFsaWduR1ElYXhpc0YnL0ZmblEpYmFzZWxpbmVGJy9GaG5RJ2NlbnRlckYnL0ZqblEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHUSV0cnVlRicvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy9GQEZbci8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zlci8lJmZyYW1lR0Zlci8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRjUvJS1lcXVhbGNvbHVtbnNHRjUvJS1kaXNwbGF5c3R5bGVHRjUvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGYnJGM0Y2LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUkjbW9HRiQ2LlExJkludmlzaWJsZVRpbWVzO0YnRjNGNi8lJmZlbmNlR0Y1LyUqc2VwYXJhdG9yR0Y1LyUpc3RyZXRjaHlHRjUvJSpzeW1tZXRyaWNHRjUvJShsYXJnZW9wR0Y1LyUubW92YWJsZWxpbWl0c0dGNS8lJ2FjY2VudEdGNS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmJ1LUZINictRks2Ni1GTjYmLUZRNigtRiM2J0YrLUYjNidGKy1JJW1zdWJHRiQ2JS1GLDYmUSJ4RicvJSdpdGFsaWNHRmhxRjMvRjdRJ2l0YWxpY0YnLUZYNiVGXW9GM0Y2LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0YzRjZGK0YzRjZGZW5GZ25GaW5GW29GXm9GZW5GZ25GaW4tRk42Ji1GUTYoLUYjNidGKy1GIzYnRistRmJ2NiVGZHYtRlg2JVEiMkYnRjNGNkZdd0YrRjNGNkYrRjNGNkZlbkZnbkZpbkZbb0Zeb0ZlbkZnbkZpbkZdcUZgcUZicUZkcUZmcUZpcUZcckZdckZgckZjckZmckZockZqckZdc0Zfc0Zhc0Zjc0Zmc0YzRjZGaHNGW3QtRl90Ni5RKSZlcXVhbHM7RidGM0Y2L0ZjdFEmdW5zZXRGJy9GZXRGYXgvRmd0RmF4L0ZpdEZheC9GW3VGYXgvRl11RmF4L0ZfdUZheC9GYXVRLDAuMjc3Nzc3OGVtRicvRmR1Rml4LUZINictRks2Ni1GTjYmLUZRNigtRiM2J0YrLUYjNiZGKy1GWDYlUSM0NUYnRjNGNkYzRjZGK0YzRjZGZW5GZ25GaW5GW29GXm9GZW5GZ25GaW4tRk42Ji1GUTYoLUYjNidGKy1GIzYmRistRlg2JVEkLTgxRidGM0Y2RjNGNkYrRjNGNkZlbkZnbkZpbkZbb0Zeb0ZlbkZnbkZpbkZdcUZgcUZicUZkcUZmcUZpcUZcckZdckZgckZjckZmckZockZqckZdc0Zfc0Zhc0Zjc0Zmc0YzRjZGaHNGW3RGKy1GMDYlUSIsRidGM0Y2LUY6NiZGPC9GQEZidUZCL0ZFUShuZXdsaW5lRidGKy1GMDYlUSV0aGVuRidGM0Y2RjlGKy1GSDYlLUZLNjYtRk42Ji1GUTYoLUYjNidGKy1GIzYnRistRiM2KUYrRmF2Rl14LUkmbWZyYWNHRiQ2KC1GWDYlUSQzNjBGJ0YzRjYtRlg2JVEiN0YnRjNGNi8lLmxpbmV0aGlja25lc3NHRl1vLyUrZGVub21hbGlnbkdGY3EvJSludW1hbGlnbkdGY3EvJSliZXZlbGxlZEdGNUYrRjNGNkYrRjNGNkYrRjNGNkZlbkZnbkZpbkZbb0Zeb0ZlbkZnbkZpbi1GTjYmLUZRNigtRiM2J0YrLUYjNidGKy1GIzYpRitGaHdGXXgtRl9cbDYoLUZYNiVRJS0yNDNGJ0YzRjYtRlg2JVEjMTRGJ0YzRjZGZ1xsRmlcbEZbXWxGXV1sRitGM0Y2RitGM0Y2RitGM0Y2RmVuRmduRmluRltvRl5vRmVuRmduRmluRl1xRmBxRmJxRmRxRmZxRmlxRlxyRl1yRmByRmNyRmZyRmhyRmpyRl1zRl9zRmFzRmNzRmZzRjNGNkYrRjNGNg==This is the reduction procedure: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right click RREF and linear solveOnce one has understood the reduction algorithm and want to simply row echelon reduce a matrix and/or solve the corresponding system of linear equations, both actions can be done directly from right-clicking on the output matrices. Note that having preloaded Student[LinearAlgebra] changes the location of these selections on the menu.
The Solvers and Forms, Row Echelon Form, Reduced menu choice will give the fully reduced form of a matrix, after which the Solvers and Forms, Linear Solve choice will perform the backwardsubstitute step to solve the corresponding linear system of equations for which it is the augmented matrix. One can also apply this last command to the original matrix without reducing it.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LinearAlgebra:-ReducedRowEchelonForm( );LinearAlgebra:-LinearSolve( );LinearAlgebra versionoptional command LinearSolve to solve Matrix equations 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First we erase the former assignment for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn so we can use it here: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 solve the linear system 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 for a column matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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Or we can introduce a vector of 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solution by augmented matrix, GaussianElimination or RowReducedEchelonForm, BackwardSubstituteIt is easy to combine submatrices together into a larger matrix, observing the rule: " | " to separate columns and " , " to separate rows: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Occasionally it is useful to use the underscore in symbols, so to turn of the subscript function of the underscore character, hit the backslash character " \134 " first.
Gaussian elimination (partial reduction to a row echelon form, without the leading 1 condition):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-Jordan elimination (full reduction to Reduced Row Echelon Form):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Backsubstitution to find the solution in either one: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 long command names, eh? Fortunately by typing the first few letters of the command and then "Control Space", you get a list of possible autocomplete choices. Try it by typing "Red", then "Control Space" in the input region:JSFHrow reduction commands (elementary row operations)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 A, multiply row 1 by the number 2 :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 A, swap row 1 and row 2 in A: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In A, add row 2 with a coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJy1JI21uR0YkNiRRIjRGJ0Y1RjVGK0Y1 to row 1: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 steps (pivoting)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 pivot command is the sequence of successive addrow operations adds multiples of the row of the entry position chosen to the other rows to make the other entries in its column 0: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Same as:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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Only the MultiplyRow command for each nonzero row remains to be done to reduce this matrix to rref form.rref: the 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 rref process is just a series of steps consisting of a pair of mulrow/pivot commands (mulrow to make the pivot entry 1, pivot to make the other entries in its column 0) down the main diagonal until a zero entry is encountered, in which case one must swap up a row from below with a nonzero entry in that column and restart the step, or if none exists, shift one entry to the right and restart the step with that entry, until one hits the lower or right edge of the matrix while pivoting in the final column.
Remember that each pivot is a sequence of addrow operations (one less than the number of rows) which makes the remaining entries of the pivot entry column equal to 0).example: no swaps, no shiftsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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example: shift necessaryWe make up an example where one has to shift right on the second pivot:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEnTWF0cml4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JS1GVzYmLUYjNiktRlc2Ji1GIzYsLUkjbW5HRiQ2JFEiMUYnRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUZebzYkUSIyRidGQEZhby1GXm82JFEiMEYnRkBGYW8tRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGY3BGXW8vJStleGVjdXRhYmxlR0ZERkBGK0ZlcEZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYW8tRlc2Ji1GIzYrRmpvRmFvRmpvRmFvRl1vRmFvRmdvRmVwRkBGQEZncEZqcEZhby1GVzYmLUYjNitGam9GYW9Gam9GYW9Gam9GYW9Gam9GZXBGQEZARmdwRmpwRmVwRkBGQEZncEZqcEZlcEZARkBGZXBGQEYrRmVwRkAtRj02LVEiO0YnRkBGQkZkb0ZHRklGS0ZNRk9GUS9GVVEsMC4yNzc3Nzc4ZW1GJ0ZlcEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2KS1GUDYmLUYjNiktSSNtbkdGJDYkUSIyRidGOS1GNjYtUSJ8Z3JGJ0Y5RjtGPi9GQUYxRkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GW28tRlk2JFEiMUYnRjlGZm4tRlk2JFEiM0YnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUTMmTGVmdEFuZ2xlQnJhY2tldDtGJy8lJmNsb3NlR1E0JlJpZ2h0QW5nbGVCcmFja2V0O0YnLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRlA2Ji1GIzYpRlhGZm4tRiM2Ji1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GXXFGXW9GY29GOUZmbkZgb0Zjb0Y5RjlGZW9GaG9GW3AtRlA2Ji1GIzYpRmBvRmZuLUYjNiZGaXBGWEZjb0Y5RmZuRl1vRmNvRjlGOUZlb0Zob0Zjb0Y5RjlGZW9GaG8tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZfcC9GTkZgcC1GLDYlUSIlRidGL0YyRmNvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEsTXVsdGlwbHlSb3dGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIkFGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMUYnRkBGaG4tSSZtZnJhY0dGJDYoLUYjNiVGXm8vJStleGVjdXRhYmxlR0ZERkAtRiM2JS1GX282JFEiMkYnRkBGZ29GQC8lLmxpbmV0aGlja25lc3NHRmFvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmJwLyUpYmV2ZWxsZWRHRkRGZ29GQEZARmdvRkBGK0Znb0ZALUY9Ni1RIjtGJ0ZARkJGW29GR0ZJRktGTUZPRlEvRlVRLDAuMjc3Nzc3OGVtRidGZ29GQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmUGl2b3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIiVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMUYnRkBGaG5GXm8vJStleGVjdXRhYmxlR0ZERkBGQEZib0ZARitGYm9GQEYrRmJvRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEsTXVsdGlwbHlSb3dGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUYsNiVRIiVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMkYnRkBGaG4tRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGaG8tSSZtZnJhY0dGJDYoLUYjNiUtRl9vNiRRIjFGJ0ZALyUrZXhlY3V0YWJsZUdGREZALUYjNiVGXm9GYnBGQC8lLmxpbmV0aGlja25lc3NHRmFwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpwLyUpYmV2ZWxsZWRHRkRGYnBGQEYrRmJwRkBGQEZicEZARitGYnBGQC1GPTYtUSI7RidGQEZCRltvRkdGSUZLRk1GT0ZRL0ZVUSwwLjI3Nzc3NzhlbUYnRmJwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmUGl2b3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIiVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMkYnRkBGaG4tRl9vNiRRIjNGJ0ZALyUrZXhlY3V0YWJsZUdGREZARkBGZW9GQEYrRmVvRkBGK0Zlb0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEsTXVsdGlwbHlSb3dGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUYsNiVRIiVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMkYnRkBGaG4tRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGaG8tSSZtZnJhY0dGJDYoLUYjNiUtRl9vNiRRIjFGJ0ZALyUrZXhlY3V0YWJsZUdGREZALUYjNiVGXm9GYnBGQC8lLmxpbmV0aGlja25lc3NHRmFwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpwLyUpYmV2ZWxsZWRHRkRGYnBGQEYrRmJwRkBGQEZicEZARitGYnBGQC1GPTYtUSI7RidGQEZCRltvRkdGSUZLRk1GT0ZRL0ZVUSwwLjI3Nzc3NzhlbUYnRmJwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmUGl2b3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIiVGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEiMkYnRkBGaG4tRl9vNiRRIjNGJ0ZALyUrZXhlY3V0YWJsZUdGREZARkBGZW9GQEYrRmVvRkBGK0Zlb0ZAexample: swap necessaryHere we have an example where a swaprow is necessary at the second pivot: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stopping rref after a given column (not available in the complete reduction commands)It is important to realize that this rref process works one column at a time moving left to right, and up to each column in the process, columns to its left never change again, so we can stop the rref process at any column if we need to. However, this functionality has been removed from the LinearAlgebra package! So we need to do a series of Pivots, Swaps and MultiplyRow operations to do a partial reduction.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Without stopping this reduction at the 3rd column, Maple assumes the last entry in the 4th column is nonzero and procedes as though this condition were satisfied, but the final result is only true when 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. If it is instead 0, we get a different result.rankThe rank of a matrix is just the number of nonzero rows in its rref reduced matrix. This infomration is available from the LinearSolve or two Reduction tutors after the matrix has been reduced by going to Properties.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjYvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjI2JkY0RjdGOi9GPVElYXV0b0YnLUYsNiVRNlJlZHVjZWRSb3dFY2hlbG9uRm9ybUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIkNGJ0ZGRkkvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GSlEnbm9ybWFsRidGV0ZURldGK0ZURlc=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inverseYou can use the superscript "^(-1)" notation for matrix inverse (if it exists!):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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMUYnRkFGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0ZBThe inverse matrix is actually obtained by row reducing the matrix augmented by the identity matrix and deleting the left half: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The InverseTutor shows you all the steps and will return the inverse matrix if you wish: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Sometimes just putting vertical delimiters around the matrix will produce the determinant (but sometimes this derails yet for some reason)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtaUdGJDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRj0vJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGQUY6Rj0=vector spaceslinear transformations (optional)The Tools Menu, Tutors selection, Linear Algebra, Linear Transforms... brings up the Linear Transformation Tutor, which illustrates in 2d and 3d cases how a square matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiQUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjVRJ25vcm1hbEYn moves points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjVRJ25vcm1hbEYn around under matrix multiplication: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUS0mUmlnaHRBcnJvdztGJy9GNVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUS1GLDYnUSJBRidGL0YyRjRGNy1GOzYuUTEmSW52aXNpYmxlVGltZXM7RicvJStleGVjdXRhYmxlR0ZCRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0ZnbkYrRj4=. The help page for the plot command explains that the default plot shows the lines along the eigenvectors of the matrix, whose significance will be discussed in the next section. Removing the number sign from the next line and executing shows the help page.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy1GLDYtUSI/RidGL0YyL0Y2RjRGOEY6RjxGPkZAL0ZDUSwwLjExMTExMTFlbUYnL0ZGRk0tRiM2Ji1JI21pR0YkNiNRIUYnLUYjNiQtRlI2JVE0TGluZWFyVHJhbnNmb3JtUGxvdEYnLyUnaXRhbGljR0Y3L0YwUSdpdGFsaWNGJ0YvRlFGL0ZRRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVE5TGluZWFyVHJhbnNmb3JtUGxvdFR1dG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2I1EhRictRiM2JkY6LUY2NigtRiM2JkY6LUknbXRhYmxlR0YkNjYtSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnRlIvJSlyb3dhbGlnbkdGPC8lLGNvbHVtbmFsaWduR0Y8LyUrZ3JvdXBhbGlnbkdGPC8lKHJvd3NwYW5HRlEvJStjb2x1bW5zcGFuR0ZRLUZKNigtRk82JFEiNEYnRlJGVEZWRlhGWkZmbkZURlZGWC1GRzYnLUZKNigtRiM2JC1GTzYkUSIyRidGUkZSRlRGVkZYRlpGZm4tRko2KC1GIzYkLUZPNiRRIjNGJ0ZSRlJGVEZWRlhGWkZmbkZURlZGWC8lJmFsaWduR1ElYXhpc0YnL0ZVUSliYXNlbGluZUYnL0ZXUSdjZW50ZXJGJy9GWVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZqcC8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZlcS8lJmZyYW1lR0ZlcS8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHUSZmYWxzZUYnLyUtZXF1YWxjb2x1bW5zR0Zfci8lLWRpc3BsYXlzdHlsZUdGX3IvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGYnFGOkZSRlIvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZpckY6RlJGOkZSRlItSSNtb0dGJDYtUSI7RidGUi8lJmZlbmNlR0Zfci8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Zfci8lKnN5bW1ldHJpY0dGX3IvJShsYXJnZW9wR0Zfci8lLm1vdmFibGVsaW1pdHNHRl9yLyUnYWNjZW50R0Zfci8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnL0ZccUZmdC8lJmRlcHRoR0ZfdS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUS1FaWdlbnZlY3RvcnNGJ0YvRjItRjY2JEY9RlJGUg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVE5TGluZWFyVHJhbnNmb3JtUGxvdFR1dG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JkYrLUYjNiZGKy1GVzYoLUYjNiZGKy1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNictSSRtdGRHRiQ2KC1GIzYlLUY9Ni1RKCZtaW51cztGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMTExMTExMWVtRicvRlVGam8tSSNtbkdGJDYkUSI1RidGQEZALyUpcm93YWxpZ25HRi4vJSxjb2x1bW5hbGlnbkdGLi8lK2dyb3VwYWxpZ25HRi4vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGaHAtRmJvNigtRl1wNiRRIjJGJ0ZARmBwRmJwRmRwRmZwRmlwRmBwRmJwRmRwLUZfbzYnLUZibzYoLUYjNiRGXXFGQEZgcEZicEZkcEZmcEZpcC1GYm82KC1GIzYlRmZvRl1xRkBGYHBGYnBGZHBGZnBGaXBGYHBGYnBGZHAvJSZhbGlnbkdRJWF4aXNGJy9GYXBRKWJhc2VsaW5lRicvRmNwUSdjZW50ZXJGJy9GZXBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0Y4LyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGZ3IvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGYnMvJSZmcmFtZUdGYnMvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZELyUtZXF1YWxjb2x1bW5zR0ZELyUtZGlzcGxheXN0eWxlR0ZELyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRl9zRitGQEZAL0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGZXRGK0ZARitGQEZARkBGK0ZALUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4yNzc3Nzc4ZW1GJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy9GaXJGUy8lJmRlcHRoR0ZpdS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLUYjNictRiw2JVEtRWlnZW52ZWN0b3JzRidGNkY5RitGVkYrRkBGK0ZAJSFHbasisA basis can be formed from any set of vectors by tossing out redundant vectors or taking new combinations which are linearly independent. Remember that Maple shuffles the order of the elements in any set each time you call up the set since the order is not supposed to be important. By using square brackets instead, Maple respects the order. In this set we have chosen two independent vectors and a third which is a linear combination of these two.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Apparently this Basis command works from left to right, eliminating any vectors which can be obtained from a linear combination of the preceding vectors along the way, picking out a linearly indepedent subset of the original set of vectors (but the set braces may shuffle the order first).One way of doing this yourself is to row reduce the matrix of column vectors (once their order is fixed) and throw away the columns corresponding to the nonleading entry columns in the reduced matrix, since these columns can always be expressed as a linear combination of the leading entry columns with coefficients which are in fact the entries in that original column. In this example all 4 vectors lie in a plane in space, and no two are proportional, so any 2 are linearly independent: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So keep the first 2 vectors and throw out the last 2, which are obvious linear combinations of the first 2: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Of course re-ordering the columns will lead to different subsets of linearly independent vectors. The rank of a matrix tells how many columns (or equivalently rows) are linearly independent.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSVSYW5rRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJBRidGNEY3Rj5GPkY+RitGPg==It is very useful to "augment" vectors together as columns in a matrix. An alternative way of doing this uses Matrix: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eigenvectors and square matrix diagonalizationEigenvalues, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEsRWlnZW52YWx1ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkFGJ0Y2RjlGQEZARkBGK0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has two distinct real eigenvalues, 1 repeated 3 times (multiplicity = 3), and 5 occuring only once (multiplicity = 1).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This is a sequence of a vector and a matrix. The vector lists the eigenvalues the matrix lists in the same order as columns a set of eigenvectors. The order of eigenvalues is random and annoyingly changes each time you re-execute the command, so to keep the correlation between the list of eigenvalues and the list of columns in the matrix one must pick them out after the Eigenvectors command has been executed using the percent " % " previous output notation or by the clever simultaneous assignment: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 we change one entry, the eigenvectors get ugly and require numerical evaluation:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYmLUkjbW5HRiQ2JFEiNEYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjpGPkY+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZBNi1RIzo9RidGPkZEL0ZIRkZGSUZLRk1GT0ZRL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRl5vLUY7NiRRIjBGJ0Y+LUZBNi1RIjpGJ0Y+RkRGXG9GSUZLRk1GT0ZRRl1vRl9vLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlUvJSZkZXB0aEdGW3AvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGKy1GQTYtUSI7RidGPkZERkdGSUZLRk1GT0ZRRlNGX29GZm8tRiw2I1EhRictRiM2NEZmcC1GIzYmRmZwLUYjNiYtRiw2JVEtRWlnZW52ZWN0b3JzRidGL0YyLUZBNi1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GREZcb0ZJRktGTUZPRlFGUy9GV0ZVLUY2NiQtRiM2JEYrRj5GPkY+RmZwRj5GY3BGZm9GZnAtRkE2LVEiJUYnRj5GREZHRklGS0ZNRk9GUUZTRl9vLUY2NiYtRiM2JC1GOzYkUSIxRidGPkY+Rj5GWUZmbkZjcEZmb0ZqcUZqcS1GNjYmLUYjNiQtRjs2JFEiMkYnRj5GPkY+RllGZm5GY3BGZm8tRiw2JVEmZXZhbGZGJ0YvRjItRjY2JC1GIzYkLUYsNiVRJCUlJUYnRi9GMkY+Rj5GZnBGPkZmcEZmb0ZmcEY+optional command: EigenPlot (2d, 3d plots)This command in 2d draws a unit circle and diameters in it which are aligned with the 2 eigenvectors if they are real, while attaching the image vector of points on the circle under matrix multiplication by the original matrix, with the tail attached to the point on the circle being multiplied. In 3d a sphere is used with similar features.
2 positive eigenvalues: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 positive, one negative eigenvalue: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One zero eigenvalue among 2 distinct real eigenvalues: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 conjugate eigenvalues: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only 1 repeated real 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If you read the help page for Eigenvectors, you see that when fewer than the maximum number of eigenvectors exist, the matrix of eigenvectors is filled with additional zero columns to make it square.optional commands: CharacteristicPolynomial, 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The vector coefficients of the parameters are the eigenvector basis for this repeated eigenvalue are: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These are the same results but possibly in a different order from the corresponding 3 columns of:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEtRWlnZW52ZWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJBRidGNkY5RkBGQEZARitGQEYrRkA=MAPLE randomly orders elements of any set, like the set of vectors which form the matrix of eigenvectors here. Each time the command is re-executed the order may change. Mabye? Each time Maple upgrades things may change.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEtRWlnZW52ZWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJBRidGNkY5RkBGQEZARitGQEYrRkA=Making a matrix of the set of all eigenvectors if sufficient in number makes a square matrix which diagonalizes the original matrix, with the diagonal values equal to the eigenvalues in the same order chosen for the eigenvectors in their matrix listing as columns: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systems of linear differential equations2d phaseplotFor a system of 2 linear homogeneous DEs in 2 dependent variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3gyRidGL0YyRjk= and an independent variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , we can use a directionfield like picture (called a phaseplot) to visualize the DE in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn plane.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[The dirgrid option enables us to change the default 20x20 gridpoint field of arrows if we wish.]LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEnREVwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Ly1GLDYlUSVkZXFzRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GVzYmLUYjNihGKy1GIzYmLUYsNiVRI3gxRidGNkY5RjwtRlc2JC1GIzYkLUYsNiVRInRGJ0Y2RjlGQEZARkBGaG4tRiM2Ji1GLDYlUSN4MkYnRjZGOUY8RmdvRkBGK0ZARkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGaG4tRiM2J0ZbcC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRl9xLUYjNiYtSSNtbkdGJDYkUSIwRidGQC1GPTYtUSMuLkYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJ0ZULUZkcTYkUSIxRidGQEZARitGQEZobi1GIzYnRmRvRltxLUYjNidGKy1GIzYlLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GT0ZRL0ZVUSwwLjExMTExMTFlbUYnLUZkcTYkUSIyRidGQEZARmdxRmpyRkBGK0ZARmhuLUYjNidGYHBGW3FGYXJGK0ZARmhuLUYjNictRiw2JVEoZGlyZ3JpZEYnRjZGOUZbcS1GVzYkLUZXNiYtRiM2Ji1GZHE2JFEjMTBGJ0ZARmhuRmpzRkBGQEZjcEZmcEZARitGQEYrRkBGQEZARitGQEYrRkA=Initial conditions then give us the curves tangent to this field of arrows starting at the initial points and proceeding in the direction of the arrows; changing the range of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tells how far to continue the curves (or even include the backwards direction, e.g. 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the straight line solutions for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSN4MkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSN4MUYnRjRGN0Y+RitGPg== and 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. These are 1-dim linear subspaces of R^2 with respective bases {LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGNA==} and {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}:
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, [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn bound, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn free], LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSN4MkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ0RidGNEY3Rj5GK0Y+, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSN4MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSN4MkYnRjRGN0Y+RitGPg== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, so x = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0YwRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ = 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;
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, [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn bound, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn free], LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSN4MkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ0RidGNEY3Rj5GK0Y+, 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, so x = 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 = 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;dsolve solutionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVElZGVxc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiYtRiM2KEYrLUYjNihGKy1GIzYoRistSSZtZnJhY0dGJDYoLUYjNiQtRj02LVEwJkRpZmZlcmVudGlhbEQ7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmJvRkAtRiM2JkYrLUYjNiVGXm8tRiw2JVEidEYnRjZGOUZARitGQC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYHAvJSliZXZlbGxlZEdGRC1GPTYuUTEmSW52aXNpYmxlVGltZXM7RicvJStleGVjdXRhYmxlR0ZERkBGQkZFRkdGSUZLRk1GT0Zhb0Zjby1GIzYmLUYsNiVRI3gxRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GT0Zhb0Zjby1GVzYkLUYjNiRGaG9GQEZARkBGK0ZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNihGK0ZqcC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRl9yLUYjNictSSNtbkdGJDYkUSIyRidGQEZlcC1GIzYmLUYsNiVRI3gyRidGNkY5Rl9xRmJxRkBGK0ZARitGQEYrRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0Zhby9GVVEsMC4zMzMzMzMzZW1GJy1GIzYoRistRiM2KEYrRmluRmVwRmdyRitGQEZmcS1GIzYoRistRiM2J0ZjckZlcEZqcEYrRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZeckZgckZhckYrRkBGK0ZARitGQEZALyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrRkBGK0ZARitGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSZpbml0c0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiM2JEYrRj5GK0Y+RistSShtZmVuY2VkR0YkNiYtRiM2Ly1GLDYlUSN4MUYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGXG8tRlc2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnRj5GPkY+RistRjs2LVEiPUYnRj5GQEZDRkVGR0ZJRktGTUZPRlItRmNvNiRRIjFGJ0Y+LUY7Ni1RIixGJ0Y+RkAvRkRGNkZFRkdGSUZLRk1GW28vRlNRLDAuMzMzMzMzM2VtRictRiM2J0YrLUYjNiYtRiw2JVEjeDJGJ0Y0RjdGaG5GXm9GPkZmb0Zib0Y+RitGK0YrRitGPkY+LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEoc29sX2dlbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Ji1GLDYlUSdkc29sdmVGJ0Y2RjktRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmluLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVElZGVxc0YnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUZcbzYmLUYjNihGKy1GIzYmLUYsNiVRI3gxRidGNkY5RmVuLUZcbzYkLUYjNiQtRiw2JVEidEYnRjZGOUZARkBGQEZjby1GIzYmLUYsNiVRI3gyRidGNkY5RmVuRmJwRkBGK0ZARkAvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRitGQEZARkBGK0ZARitGQEYrRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiZGKy1GIzYnLUYsNiVRKnNvbF9pbml0c0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjxRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtRiM2Ji1GLDYlUSdkc29sdmVGJ0Y4RjstRj82LVEwJkFwcGx5RnVuY3Rpb247RidGQkZERkdGSUZLRk1GT0ZRL0ZUUSYwLjBlbUYnL0ZXRltvLUkobWZlbmNlZEdGJDYkLUYjNihGKy1GIzYmLUYsNiVRJWRlcXNGJ0Y4RjstRj82LVEmJmN1cDtGJ0ZCRkQvRkhGOkZJRktGTUZPRlFGam4vRldRLDAuMzMzMzMzM2VtRictRiw2JVEmaW5pdHNGJ0Y4RjtGQi1GPzYtUSIsRidGQkZERmpvRklGS0ZNRk9GUUZqbkZbcC1GXm82Ji1GIzYoRistRiM2Ji1GLDYlUSN4MUYnRjhGO0Znbi1GXm82JC1GIzYkLUYsNiVRInRGJ0Y4RjtGQkZCRkJGYHAtRiM2Ji1GLDYlUSN4MkYnRjhGO0ZnbkZccUZCRitGQkZCLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrRkJGQkZCRitGQkYrRkJGK0ZCRitGQg==For 2 or 3 variables, we can plot the solution curves.
To plot these in a specific order to distinguish the two curves in this case (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is red, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is blue):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSVwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2MEYrLUYjNiYtRiw2JVElc3Vic0YnRjRGN0Y6LUZVNiQtRiM2KEYrLUYjNiQtRiw2JVEqc29sX2luaXRzRidGNEY3Rj4tRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnLUZVNiYtRiM2KEYrLUYjNiYtRiw2JVEjeDFGJ0Y0RjdGOi1GVTYkLUYjNiQtRiw2JVEidEYnRjRGN0Y+Rj5GPkZhby1GIzYmLUYsNiVRI3gyRidGNEY3RjpGYHBGPkYrRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0YrRj5GPkY+RmFvRmRwLUY7Ni1RIj1GJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGZnEtSSNtbkdGJDYkUSIwRidGPi1GOzYtUSMuLkYnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJ0ZSLUZpcTYkUSIxRidGPkZhby1GLDYlUSZjb2xvckYnRjRGN0ZicS1GVTYmLUYjNiYtRiw2JVEkcmVkRidGNEY3RmFvLUYsNiVRJWJsdWVGJ0Y0RjdGPkY+RlxxRl9xRitGPkY+Rj5GK0Y+eigenvector solution (real, distinct eigenvalues)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This can be expressed in matrix form as an arbitrary linear combination of two basic solutions: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The exponential rate coefficients are the eigenvalues and the vectors are the corresponding eigenvectors of the coefficient matrix of the system of linear differential equations: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 form of the above linear system of linear first order differential equations (not useful in terms of the MAPLE commands):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNihGKy1GIzYmLUYsNiVRJG1hcEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y8USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVElZGlmZkYnRjhGOy1GPzYtUSIsRidGQkZEL0ZIRjpGSUZLRk1GT0ZRRlMvRldRLDAuMzMzMzMzM2VtRictRiw2JVEiWEYnRjhGO0Zqbi1GLDYlUSJ0RidGOEY7RkJGQkZCLUY/Ni1RIj1GJ0ZCRkRGR0ZJRktGTUZPRlEvRlRRLDAuMjc3Nzc3OGVtRicvRldGam8tRiM2Ji1GLDYlUSJBRidGOEY7LUY/Ni1RIi5GJ0ZCRkRGR0ZJRktGTUZPRlFGU0ZWRmBvRkJGK0ZCRitGQkYrRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEtRWlnZW52ZWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJBRidGNkY5RkBGQEZARitGQEYrRkA=This tells us that A has two eigenvalues LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiM0YnRjVGNS1GMjYtUSIsRidGNUY4L0Y8USV0cnVlRidGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLUSwwLjMzMzMzMzNlbUYnLUZNNiRRIjJGJ0Y1RjU=, which occur as single roots of the characteristic equation (not repeated), and each of which has 1 independent eigenvector. The two eigenvectors together form a basis of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GOVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPw==, and the corresponding basis changing matrix is just obtained by augmenting them into the columns of a matrix, given as the second output of the Eigenvectors command: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This is the sense in which B is said to "diagonalize" the matrix A.The general solution is an arbitrary linear combination of the products of the exponentials with the eigenvalues as the rate coefficients and the corresponding eigenvectors: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSZYX2dlbkYnRjZGOUZARitGQEYrRkA=The initial value problem amounts to solving 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 for the unknown coefficients 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in terms of the specified initial values of the variables: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One must now backsubstitute these values into the general solution to finish the problem. Notice that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjQzFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI0MyRidGL0YyRjk= may or may not coincide with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSRfQzJGJ0YvRjJGOQ== of MAPLE's dsolve solution due to the random switching of the order of the eigenvalues.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[in progress?? Maybe not. This may be the end. Does anyone ever read this far anyway?]JSFH