<s>[INST] Similarity of Linear Transformations in $\mathbb{R}^4$ Define $\mathbb{R}^4 \rightarrow \mathbb{R}^4$ by $T(x,y,z,w) = (x+z, y-z,z,w-z)$. 
Let $$B = \{(1,2,3,4), (1,2,3,0), (1,2,0,0), (1,0,0,0)\}$$ 
$$B' = \{(1,0,-1,0), (0,2,3,0), (1,0,0,0), (1,1,1,1)\}$$
Find the matrix $[T]_B$ of the linear transformation $T$ with respect to basis B, the matrix $[T]_{B'}$  of the linear transformation $T$ with respect to the basis $B'$, the transition matrix $P$ from $B$ to $B'$ and verify that $[T]_B$ and $[T]_{B'}$ are similar matrices. 


*

*I can easily find the transition matrix. Where I'm having issues is finding $[T]_B$ and $[T]_{B'}$

*Can I simply plug the vectors into the operation? 
$$T \begin{pmatrix} \begin{bmatrix} 1\\2\\3\\4 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 1+3\\2-3\\3\\4-3 \end{bmatrix} = \begin{bmatrix} 4\\-1\\3\\1 \end{bmatrix} $$
Repeat with all other vectors...
$$[T]_B = \begin{bmatrix} 4&4&1&1\\-1&-1&2&0\\3&3&0&0\\1&-3&0&0 \end{bmatrix}$$
Similar process with $[T]_{B'}$
Once I find transition matrix $P$, checking the similarity of $B$ and $B'$ is a simply a matter of checking if:
$$P^{-1}BP = B'$$
Is this correct? Are there any other requirements that I have to check for similarity between matrices? 
 [/INST] You have calculated is the matrix $[T]_{B,E}$, where $E$ is the standard basis.
In general if $\mathscr{A}, \mathscr{B}$ are two bases then, the n th column of $[T]_{\mathscr{A}, \mathscr{B}}$ is $ \begin{bmatrix} \lambda_1\\ \vdots\\ \lambda_n \end{bmatrix}$, 
where $T(a_n) = \Sigma_ i \lambda_i b_i $ and $a_i, \in \mathscr{A}$,$b_i, \in \mathscr{B}$.
So for $[T]_B = [T]_{B,B}$  compute:
$$
\begin{align}
T \begin{pmatrix} \begin{bmatrix} 1\\2\\3\\4 \end{bmatrix} \end{pmatrix} & = \begin{bmatrix} 4\\-1\\3\\1 \end{bmatrix} = \frac{1}{4} \begin{bmatrix} 1\\2\\3\\4 \end{bmatrix} + \frac{3}{4} \begin{bmatrix} 1\\2\\3\\0 \end{bmatrix} + \frac{-3}{2} \begin{bmatrix} 1\\2\\0\\0 \end{bmatrix} + \frac{9}{2} \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix}
\end{align}
$$
Thus first column of $[T]_B$ is $ \begin{bmatrix} \frac{1}{4}\\ \frac{3}{4}\\ \frac{-3}{2}\\ \frac{9}{2}\\  \end{bmatrix}$.
Repeating for other basis vectors will yield $[T]_B$ . 
Your calculation for verifying similar matrices is correct.
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