Standard Matrices of Linear Transformations

Question

Find the standard matrix of the linear transformations T that act as follows:

(a) T(x, y) = (-4x + 6y, -x - 5y)

(b) T(x y,) = (-2x - y, 2x + 3y, -4x + y)

Answer 1

(a) T(1,0) = (-4·1 + 6·0, -1 - 5 · 0) = (-4, -1)

T(0,1) = (-4·0 + 6·1, -0 - 5·1) = (6, -5)

Because Standard Matrix of T is 2×2 matrix with 1st column is (-4, -1) and 2nd column is (6, -5)

(b) T(1, 0) = (-2·1 - 0, 2·1 + 3·0, -4·1 + 0) = (-2, 2, -4)

T(0, 1) = (-2·0 - 1, 2·0 + 3·1, -4·0 + 1) = (-1, 3, 1)

Because Standard Matrix of T is 3×2 matrix wist first column(-2,2,-4) and 2nd column(-1,3,1)