Let V and W be vector spaces, and let L: V -> W be a linear transformation between them. A basis for V is E = {$v_1$,...,$v_5$}. A basis for W is F = {$w_1$,...,$w_4$}. On the basis vectors the linear transformation does the following:

$L(v_1) = 3w_1 + 2w_2 + w_4$

$L(v_2) = 2w_1 - 2w_2 + 3w_4$

$L(v_3) = -2w_1 - w_3 - w_4$

$L(v_4) = w_1 + 2w_2 - 3w_3 + w_4$

$L(v_5) = -3w_2 + 4w_4$

Find the matrix representation for the linear transformation with respect to basis E and basis F.

I have this from my notes: $[L(v)]_F = A[v]_E$ and some examples that don't seem to apply to this problem. I really don't understand how to do this problem and I've been going to YouTube and many other help sites and they don't seem to help much.

Also, I wasn't sure how to do vector notation, but all the $v$'s and $w$'s should have the normal vector arrows above them..


By definition the matrix is

$T_{F,E} = \big[(L(v_1))_F,...,(L(v_5))_F\big]$

and you know every $(L(v_i))_F$ so just put them in columns and form the matrix.

  • $\begingroup$ I had gotten as far as finding the matrix A = [3 2 -2 1 0; 2 -2 0 2 -3; 0 0 -1 -3 0; 1 3 -1 1 4] already. But it's asking for the entire transformation represented as a matrix equation if I'm reading that correctly. I just don't know where/how to situate the other vectors around matrix A. $\endgroup$ – UnworthyToast Apr 30 '14 at 3:05

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