find the standard matrix for the linear operator t defined by the formula T(x,y,z)=(x-2y ,2x+y)

Question

find the standard matrix for the linear operator t defined by the formula

T (x, y, z) = (x - 2 y, 2 x + y)

Answer 1

Here the linear transformation is given by

T (x, y, z) = (x - 2 y, 2 x + y) .

The standard basis of

R^{3} i s B 2 = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

T (1, 0, 0) = (1, 2) = 1 (1, 0) + 2 (0, 1)

T (0, 1, 0) = (- 2, 1) = - 2 (1, 0) + 1 (0, 1)

T (0, 0, 1) = (0, 0) = 0 (1, 0) + 0 (0, 1) .

Therefore the standard matrix of T is

[T] [B 2, B 3] = [1, 2, - 2, 1, 0, 0] .

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