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Unformatted text preview: These notes closely follow the presentation of the material given in David C. Lays textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class 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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