Suppose linear transformation L : R2 ?R3 is de?ned by:

Question

Suppose linear transformation L : R2 →R3 is de?ned by:

 

L [ x ]    = [ 3x-y]

   [ y]        [ x+y]

                [    -x]

 

find the Matrix of L:

 

1) with respect to the standard bases in R2 and R3;

2) with respect to bases B = {u1,u2} and B' = {v1,v2,v3}, where

 

u1=(-1,1), u2=(1,2), v1=(2,-2,1), v2=(1,0,-1), v3=(5,-3,1) .

 

 

Answer 1

Sammy,

 

The matrix of L w.r.t. the standard basis is gotten by simply taking the coefficients of x as one column, and the coefficients of y as the next column.

 

M = (3  -1; 1  1; -1  0)

 

To get the matrix w.r.t. the new bases of R² and R³ respectively, it is necessary to write down the transition matrix from the new basis to the old basis in R² followed by the transition matrix from the old basis to the new basis in R³.

 

The transition matrix from the new to the old basis in R² is 

 

P = (-1  1; 1  2)

 

while the transition from the old to the new basis in R³ is 

 

Q = (2  1  5; -2  0  -3; 1  -1  1)^-1

   = (1/3)⋅(-3  -6  -3; -1  -3  -4; 2  3  2)

 

Now, to find the matrix for L, simply note that it will be

 

L = QMP = (3  -6; 0  -2; -2  3)

 

I hope this is clear.

 

Regards,

Hassan