Let T : R3 -- R3 be a linear transformation defined by T(x, y, z) =(x + y -z, x + y + z, y - z)Then, the matrix of the linear transformation T with respect to the ordered basis B = {(0,1,0), (0,0,1), (1,0, 0)| of R3 isa)b)c)d)None of theseCorrect answer is option 'C'. Can you explain this answer?

Question

Let T : R3 --> R3 be a linear transformation defined by T(x, y, z) =(x + y -z, x + y + z, y - z)Then, the matrix of the linear transformation T with respect to the ordered basis B = {(0,1,0), (0,0,1), (1,0, 0)| of R3 isa)b)c)d)None of theseCorrect answer is option 'C'. Can you explain this answer?

Accepted Answer

Let T : R³ ---> R³ be a linear transformation defined by
T(x,y,z) = (x + y - z , x + y + z, y - z)
We need to find the matrix of T with respect to the ordered basis B = {(0, 1,0), (0,0, 1), (1, 0,0)} of R³.
Now
T(0,1,0) = (1,1,1)
= 1(0, 1, 0) + 1(0, 0, 1) + 1(0, 0, 0)
T(0, 0, 1) = (-1, 1, -1)
= 1(0, 1, 0 ) - 1(0, 0 , 1) - 1( 1, 0 , 0)
T(1, 0, 0) =(1,1,0)
= 1(0,1,0) + 0(0,0,1) + 1(1,0,0)
Therefore, the matrix of T with respect to the ordered basis
B = {(0, 1, 0), (0, 0, 1), (1, 0, 0)} is

Answer

T

:

R^{3} \to R^{3}, T (x, y, z) = (x 2, x y, y2)

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