Solving Matrix EquationsWorksheet by Michael K. May S.J., revised by Russell Blyth. Revised by Harry S. Mills into Document Mode.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The basic objectives are:Learn to use Maple to multiply matrices and vectors.Learn to use Maple to solve Matrix-Vector equationsComplete an extended exercise.
<Text-field style="Heading 1" layout="Heading 1">Part 1: Multiplying a matrix and a vector</Text-field>We want to change our point of view. So far we have been focused on solving systems of linear equations and have manipulated matrices to perform that task. In this section we consider the equivalent matrix equation and how to solve it directly. Before we can talk about solving matrix equations we need to be able to multiply a matrix and an appropriately sized vector. We start by defining a matrix A and a vector X of appropriate size so that the product AX is defined. (The product AX of an m by n matrix A with the vector X is only defined if X is an n-vector.)Define A by entering its rows:NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNictSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y7LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtRiQ2WC1GLTY5USJBRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tSSdtc3BhY2VHRiU2Ji8lJ2hlaWdodEdRJzAuMH5leEYoLyUmd2lkdGhHUScwLjV+ZW1GKC8lJmRlcHRoR0Zoby8lKmxpbmVicmVha0dRJWF1dG9GKC1JI21vR0YlNjNRIzo9RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EvdGhpY2ttYXRoc3BhY2VGKC8lJ3JzcGFjZUdGXnEvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUwZm9udF9zdHlsZV9uYW1lR0ZYLyUlc2l6ZUdGNS8lK2ZvcmVncm91bmRHRkQvJStiYWNrZ3JvdW5kR0ZHRmNvLUYtNjlRJ01hdHJpeEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZicDYzUSIoRigvRmZwUSdwcmVmaXhGKC9GaXBGO0ZqcC9GXXFRLnRoaW5tYXRoc3BhY2VGKC9GYHFGY3MvRmJxRjtGY3FGZXFGaHFGW3JGXXJGX3JGYXJGY3JGZXJGZ3ItSSNtbkdGJTY5USIzRihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ24vRmpuUSdub3JtYWxGKEZcby1GYnA2M1EiLEYoRmVwRmhwL0ZbcUY7L0ZdcVEkMGVtRigvRmBxUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGKEZhcUZjcUZlcUZocUZbckZdckZfckZhckZjckZlckZnci1GZ3M2OVEiNEYoRjBGM0Y2RmpzRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZbdEZcb0ZddC1GYnA2M1EiW0YoRl9zRmFzRmpwRmJzRmRzRmVzRmNxRmVxRmhxRltyRl1yRl9yRmFyRmNyRmVyRmdyRmh0LUZnczY5RmpxRjBGM0Y2RmpzRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZbdEZcb0ZddEZjby1GZ3M2OVEiMkYoRjBGM0Y2RmpzRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZbdEZcb0ZddEZjb0ZbdUZddEZjb0Zmcy1GYnA2M1EiXUYoL0ZmcFEocG9zdGZpeEYoRmFzRmpwRmJzL0ZgcVEydmVyeXRoaW5tYXRoc3BhY2VGKEZlc0ZjcUZlcUZocUZbckZdckZfckZhckZjckZlckZnckZddEZjb0ZodC1GYnA2M1EiLUYoRmVwRmhwRmpwL0ZdcVEwbWVkaXVtbWF0aHNwYWNlRigvRmBxRlt2RmFxRmNxRmVxRmhxRltyRl1yRl9yRmFyRmNyRmVyRmdyLUZnczY5USI1RihGMEYzRjZGanNGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRlt0RlxvRl10RmNvLUZnczY5USI3RihGMEYzRjZGanNGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRlt0RlxvRl10RmNvRl11Rl10RmNvRl11RmB1Rl10Rmh0RmNvLUZnczY5USMxM0YoRjBGM0Y2RmpzRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZbdEZcb0ZddEZjb0ZldEZddEZjb0ZldEZddEZjb0Zmc0ZgdUZgdS1GYnA2M1EiKUYoRmN1RmFzRmpwRmJzRmV1RmVzRmNxRmVxRmhxRltyRl1yRl9yRmFyRmNyRmVyRmdyRiwtRmRvNiZGZm8vRmpvUScwLjB+ZW1GKEZccC9GX3BRKG5ld2xpbmVGKEYsNyM+SSJBR0YoLUknTWF0cml4R0YoNiUiIiQiIiU3JTcmIiIiIiIjRml3RmV3NyYhIiYiIihGandGanc3JiIjOEZmd0Zmd0Zldw==print();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print();We can create the same matrix and vector using the angle bracket constructor. Now we're thinking in columns: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print();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print();Maple uses a period "." to denote matrix (non-commutative) multiplication, so to compute the product AX, we write...NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiktSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtSSdtc3BhY2VHRiU2Ji8lJ2hlaWdodEdRJzAuMH5leEYoLyUmd2lkdGhHUScwLjB+ZW1GKC8lJmRlcHRoR0Zjby8lKmxpbmVicmVha0dRKG5ld2xpbmVGKC1GLTY5USJCRihGMEYzRjZGOUY8Rj5GQEZCRkVGSC9GS0Y7RkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tSSNtb0dGJTYzUSM6PUYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRigvJSdyc3BhY2VHRl1xLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1GLTY5USJBRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZfcEZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZhcDYzUSIuRihGZHBGZ3BGaXAvRlxxUSQwZW1GKC9GX3FGX3NGYHFGYnFGZHFGZ3FGanFGXHJGXnJGYHJGYnJGZHJGZnItRi02OVEiWEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGX3BGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcbzcjPkkiQkdGKC1JMGRlbGF5RG90UHJvZHVjdEdGJTYkSSJBR0YoSSJYR0Yoprint();print();Maple also lets you use commands for the operations. The command to multiply a matrix followed by a vector is MatrixVectorMultiply. There are similar commands for a vector and a matrix as well as for the product of two matrices.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();The product AX can be thought of as a linear combination of the columns of A with the scalars used being the entries of X. If we compute this linear combination, we should get the same result.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGOi8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy8lKm1hdGhjb2xvckdGQy8lL21hdGhiYWNrZ3JvdW5kR0ZGLyUrZm9udGZhbWlseUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSltYXRoc2l6ZUdGNA==NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y7LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtRiQ2US1GLTY5USdDb2x1bW5GKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1JI21vR0YlNjNRIihGKC8lJWZvcm1HUSdwcmVmaXhGKC8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EudGhpbm1hdGhzcGFjZUYoLyUncnNwYWNlR0ZgcC8lKXN0cmV0Y2h5R0Y7LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRi02OVEiQUYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZkbzYzUSIsRigvRmhvUSZpbmZpeEYoL0ZbcEY4L0ZdcEY7L0ZfcFEkMGVtRigvRmJwUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGKC9GZHBGOEZlcEZncEZqcEZdcUZfcUZhcUZjcUZlcUZncUZpcS1JI21uR0YlNjlGXHFGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ24vRmpuUSdub3JtYWxGKEZcby1GZG82M1EiKUYoL0Zob1EocG9zdGZpeEYoRmpvRlxwRl5wL0ZicFEydmVyeXRoaW5tYXRoc3BhY2VGKEZjcEZlcEZncEZqcEZdcUZfcUZhcUZjcUZlcUZncUZpcS1GZG82M1EiKkYoRmFyRmNyRlxwRl5wRmFwRmlyRmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxLUYtNjlRIlhGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1GZG82M1EiW0YoRmdvRmpvRlxwRl5wRmFwRmNwRmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxRmpyLUZkbzYzUSJdRihGY3NGam9GXHBGXnBGZXNGY3BGZXBGZ3BGanBGXXFGX3FGYXFGY3FGZXFGZ3FGaXEtRmRvNjNRIitGKEZhckZjckZccC9GX3BRMG1lZGl1bW1hdGhzcGFjZUYoL0ZicEZndEZpckZlcEZncEZqcEZdcUZfcUZhcUZjcUZlcUZncUZpcUZgb0Zjb0ZbckZeci1GW3M2OVEiMkYoRjBGM0Y2Rl1zRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZec0Zcb0Zgc0Znc0Zqc0ZddEZpdEZgdEZjdEZgb0Zjb0ZbckZeci1GW3M2OVEiM0YoRjBGM0Y2Rl1zRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZec0Zcb0Zgc0Znc0Zqc0ZddEZcdUZgdEZjdEZgb0Zjb0ZbckZeci1GW3M2OVEiNEYoRjBGM0Y2Rl1zRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZec0Zcb0Zgc0Znc0Zqc0ZddEZfdUZgdEYsNyMsKiomLUknQ29sdW1uRzYkL0YnSS5MaW5lYXJBbGdlYnJhRzYkJSpwcm90ZWN0ZWRHRipGW3Y2JEkiQUdGKCIiIkZediZJIlhHRig2I0ZedkZedkZediomLUZmdTYkRl12IiIjRl52JkZgdjYjRmV2Rl52Rl52KiYtRmZ1NiRGXXYiIiRGXnYmRmB2NiNGW3dGXnZGXnYqJi1GZnU2JEZddiIiJUZediZGYHY2I0Zhd0ZedkZedg==print();Note that multiplication of a matrix by a vector is not commutative. In this case, X.A is not even defined: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 we want to put two matrices (or a matrix and a vector) together into a larger matrix. We can do this with the angle bracket constructor, using a comma to put matrices on top of each other and a vertical bar to put them next to each other.Augment A by a column vector: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print();Augment A by a row vector: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();Use the commands RandomMatrix and RandomVector to create a random 4 by 5 matrix named Matrix01 and a 5-vector named Vector01. Compute the product Matrix01.Vector01, and assign this product the name Vector02.Augment Matrix01 by Vector02 to create a 4 by 6 matrix. THEN stack Vector01 on top of Matrix01 to create a 5 by 5 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();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
<Text-field style="Heading 2" layout="Heading 2">Exercises:</Text-field>Use the commands RandomMatrix and RandomVector to create a random 4 by 5 matrix named Matrix01 and a 5-vector named Vector01. Compute the product Matrix01.Vector01, and assign this product the name Vector02.Augment Matrix01 by Vector02 to create a 4 by 6 matrix. THEN stack Vector01 on top of Matrix01 to create a 5 by 5 matrix.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 1) Use the commands RandomMatrix and RandomVector to create a random 4 by 5 matrix named Mat1 and a 5-vector named Vec1. Compute the product Mat1 . Vec1, assigned to a 4-vector named Vec2.RandomMatrix(4,5,generator=1..9); 2) Augment Mat1 by Vec2 to create a 4 by 6 matrix. (Read carefully:) Stack Vec1 on top of Mat1 to create a 5 by 5 matrix.
<Text-field style="Heading 1" layout="Heading 1">Part 2: Solving a matrix-vector equation</Text-field>Having computed B = AX, we want to shift the focus: given the matrix A and the vector B, solve for X in the matrix equation AX = B.The first method is to make an augmented matrix from A and B, then use Gaussian elimination to produce a matrix in echelon form. We then use back substitution to find the general solution to the matrix equation.C1 := <A|B>;
C1a := GaussianElimination(C1);
gensol1 := BackwardSubstitute(C1a,free=t);Here 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 is a parameter that can take any value. To get back to our original X, we need to set 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 equal to 4.eval(gensol1,t[1]=4);A slight variation of the method above is to perform Gauss-Jordan elimination before back substitution.C1b := ReducedRowEchelonForm(C1);
gensol2 := BackwardSubstitute(C1b);A third option is to use the linsolve command that we saw in the previous worksheet.gensol3 := LinearSolve(A,B);A fourth method notes that the general solution to the equation AX = B is any vector of the form tv+nv, where tv is a particular solution to the equation (a translation vector), and nv is any vector in the null space of A. The Maple command nullspace(A) returns a basis for the nullspace of A. (A basis of a vector space is a spanning set of minimal size.)NullSpace(A);
<Text-field style="Heading 2" layout="Heading 2">Exercises:</Text-field> 3) Find a basis of the null space of the matrix Mat1 you created above. 4) Solve the matrix-vector equation (Mat1)(X) = Vec2 by reducing an augmented matrix.5) Solve the matrix-vector equation (Mat1)(X) = Vec2 by using the linsolve command.
<Text-field style="Heading 1" layout="Heading 1">Part 3: An Extended exercise:</Text-field>For these exercises we will work with a fixed matrix A and a fixed vector B.Let A = 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 and B = 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.Exercises: 6) Compute the rank of A and determine the number of free variables in the system AX = 0. Find a basis for the nullspace of A. (Name the vectors of your basis W1, W2, ...) How could you have predicted the size of the basis? (Use NullSpace command)NullSpace By inspection, find a vector X0 such that (A)(X0)=B. Show that any vector of the form X0 + a1*W1 + a2*W2 + ... is a solution to the equation AX = B.