Question

let T be a linear map on R³ Such that T(1,0,0)=(1,1,1),T(0,1,0)=(0,3,5),T(0,0,1)=(2,2,2)

a)find matrix A representing T with respect to the usual basis of R³

b)find matrix B representing T with respect to the basis S={(1,2,3),(2,3,4),(-1,0,1)}

Answer 1

Given that T(1,0,0) = (1,1,1), T(0,1,0) =(0,3,5) and T(0,0,1) =(2,2,2).

a) So matrix A with respect to the usual basis is

$A = \begin{bmatrix} 1&0&2\\1&3&2\\ 1&5&2 \end{bmatrix}$

b) In order to get the matrix with respect to the basis S, we need to get the linear transformation that led us to matrix A.

$T \begin{bmatrix} a \\b \\c \end{bmatrix} = \begin{bmatrix} 1&0&2\\1&3&2\\ 1&5&2 \end{bmatrix} \begin{bmatrix} a \\b \\c \end{bmatrix}$

i.e $T(a,b,c) = (a+2c,a+3b+2c,a+5b+2c)$

$T(1,2,3) = (1+2(3), 1+3(2)+2(3), 1+5(2)+2(3)) = (1+6,1+6+6,1+10+6) = (7,13,17)$

$T(2,3,4) = (2+2(4),2+3(3)+2(4),2+5(3)+2(4)) = (2+8, 2+9+8,2+15+8) = (10,19,25)$

$T(-1,0,1) =(-1+2(1),-1+3(0)+2(1),-1+5(0)+2(1)) = (-1+2,-1+2,-1+2) =(1,1,1)$

So, the matrix B representing T with respect to the basis S is

$B= \begin{bmatrix} 7&10&1 \\13&19&1\\17&25&1 \end{bmatrix}$