lecture_8

# lecture_8 - MA 35100 LECTURE NOTES FRIDAY JANUARY 29 Matrix...

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MA 35100 LECTURE NOTES: FRIDAY, JANUARY 29 Matrix of a Linear Transformation In the previous lecture, we saw that given a system of n equations in m unknowns, we could express it in the form A ~x = ~ b in terms of an n × m matrix A and vectors ~x R m and ~ b R n . Similarly, a linear transformation T : R m R n can be expressed in the form T ( ~x ) = A ~x in terms of an n × m matrix and vector ~x R m . Hence, a system of equations can be expressed as a linear transformation, where T ( ~x ) = ~ b . Conversely, say that we have a linear transformation T : R m R n . We will discuss how to find the n × m matrix A . First we begin with an example to gain some intuition. Say that we have the transformation T : R 3 R 3 given by T ( ~x ) = A ~x in terms of the 3 × 3 matrix A = 1 2 3 4 5 6 7 8 9 . We want to compute T 1 0 0 , T 0 1 0 , and T 0 0 1 . To do so, write A = ~v 1 ~v 2 ~v 3 in terms of the columns of A . By definition we have the product A ~x = x 1 ~v 1 + x 2 ~v 2 + x 3 ~v 3 . Hence we compute T 1 0 0

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