Matrix representation of linear transformation with respect to a basis $$b1=(−1,0,0,1,4)$$ $$b2=(0,1,−1,0,0)$$ $$b3=(−2,2,−1,3,8)$$ $$b4=(3,−6,4,−4,−10)$$ $$b5=(−5,12,−8,7,17)$$
A linear transformation $T:\mathbb{R}^5→\mathbb{R}^5$ is given by:
$$T(u,v,w,x,y)=(4u+4v+5w+x+5y,2u−2v+3w−4x+5y,w,x,3u+5v+4w−4x−3y).$$
I calculated the matrix representation of T with respect to S as follows (which is correct):
$$T_S = \begin{bmatrix}4&4&5&1&5 \\ 2&-2&3&-4&5 \\ 0&0&1&0&0 \\ 0&0&0&1&0  \\ 3&5&4&-4&-3\end{bmatrix}$$
now I have to find the matrix representation $[T]_B$ of T with respect to B.
my understanding is that I should get it by $T_S * B$. However, that doesn't give the correct answer nor does $B * T_S$.
I even did hand calculations by transforming each column of B manually (which is the same as $T_S * B$ but to no avail.
Can someone please tell me where I am going wrong?
 A: Let $C$ be the matrix whose columns are the vectors of $B$. Then $[T]_B=C^{-1}.[T]_S.C$.
A: I like to think about this in terms of “input” and “output” bases of the matrix: it expects as inputs vectors expressed relative to some basis and outputs vectors expressed in some, perhaps different, basis. To emphasize this, I prefer a notation that makes these two bases explicit: instead of $T_{\mathcal B}$ I would write $[T]_{\mathcal B}^{\mathcal B}$. If the input and output bases were different, say, $\mathcal B$ and $\mathcal C$, respectively, this would be denoted by $[T]_{\mathcal C}^{\mathcal B}$. The change-of-basis matrix from $\mathcal B$ to $\mathcal C$ is just the matrix of the identity map with appropriate input and output bases: $[\operatorname{id}]_{\mathcal C}^{\mathcal B}$.
The first matrix that you found has the standard basis for both its input and output. The matrix that you’re tasked with finding wants $B$ for both its input and output. What you did was to compute $[T]_S^S[\operatorname{id}]_S^B = [T]_S^B$. That is to say, the resulting matrix accepts $B$-vectors, but still outputs $S$-vectors: you’ve only converted one side of the process. Similarly, when you tried multiplying on the left instead, you converted only the output side (and incorrectly to boot since the correct change-of-basis matrix there is $B^{-1}$). You need to convert both the input and output sides, i.e., you need $[T]_B^B=[\operatorname{id}]_B^S[T]_S^S[\operatorname{id}]_S^B$, which in your notation would be $B^{-1}T_SB$.
