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Unformatted text preview: These notes closely follow the presentation of the material given in David C. Lay’s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein. The Matrix of a Linear Transformation We have seen that any matrix transformation x A x is a linear transformation. The converse is also true. Specifically, if T : n m is a linear transformation, then there is a unique m n matrix, A , such that T x A x for all x n . The next example illustrates how to find this matrix. Example Let T : 2 3 be the linear transformation defined by T x 1 x 2 x 1 2 x 2 x 2 3 x 1 5 x 2 . Find the matrix, A, such that T x A x for all x 2 . Solution The key here is to use the two “standard basis” vectors for 2 . These are the vectors e 1 1 and e 2 1 . Any vector x x 1 x 2 2 is a linear combination of e 1 and e 2 because x x 1 x 2 x 1 x 2 x 1 e 1 x 2 e 2 ....
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 Spring '08
 KLEMES
 Math, Linear Algebra, Algebra, linear transformation

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