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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?

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  • $\begingroup$ If my answer below is unclear then please let me know so I can improve it $\endgroup$ – Bysshed Apr 9 '16 at 1:49
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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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