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?

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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.