Rewriting the matrix associated with a linear transformation in another basis

Question

Let $L : \mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation such that its matrix with respect to the standard basis is: $$[L] = \begin{bmatrix}1 & 0 & 1 \\ -1 & -1 & 0 \\0&1&-1\end{bmatrix}$$ Consider the two following bases of $\mathbb{R}^3$: $$\beta = \left(\begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix},\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix},\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right)$$ $$\gamma = \left(\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix},\begin{bmatrix}0 \\ 1 \\ 2\end{bmatrix},\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right)$$ Find the matrix associated with the transformation with respect to $\beta$ in input and to $\gamma$ in output.

I seem to have some troubles with this kind of exercises. I have been explained a way to do it through multiplying $3$ matrices, but I can't seem to quite remember it every time I do an exercise. This is what I have tried:

The first thing is to rewrite the vectors in $\beta$ as linear combinations of those of the standard basis. Let's enumerate the vectors in $\beta$ and $\gamma$ with an index that goes from $1$ to $3$. Also, $e_1, e_2, e_3$ are the vectors of the standard basis of $\mathbb{R}^3$.

We observe that $\beta_1 = e_1 + e_2, \beta_2 = e_2+e_3, \beta_3 = e_3$, therefore the matrix that uses $\beta$ as the input basis and outputs vectors in the standard basis will be: $$[L]^{\beta}= \begin{bmatrix}1 & 1 & 1 \\ -2 & -1 & 0 \\0&0&-1\end{bmatrix}$$

Now, if my understanding is correct, I need to write the vectors that make up the columns of that matrix as linear combinations of vectors in $\gamma$.

Using Gauss' algorithm, we find that: $$\left[\begin{array}{ccc|ccc}1&0&0&1 & 1 & 1 \\ 2&1&0&-2 & -1 & 0 \\3&2&1&0&0&-1\end{array}\right]\rightarrow \left[\begin{array}{ccc|ccc}1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & -4 & -3 & -2 \\0 & 0 & 1 & 5 & 6 & 2\end{array}\right]$$

Which finally means that: $$[L]^{\beta}_{\gamma} = \left[\begin{array} & 1 & 1 & 1 \\ -4 & -3 & -2 \\5 & 6 & 2\end{array}\right]$$

Is my reasoning correct? Did I make any mistakes?

Answer 1

I would make it conceptually simpler:

Denote $P_\beta$ and $P_\gamma$ the change of basis matrices from the standard basis to the bases $\beta$ and $gamma$ respectively. These are the matrices with columns the coordinates of the vectors in $\beta$ and $gamma$ w.r.t. the standard basis.

Also, if $X$ is a column vector in the standard basis, denote $X_\beta$ and $X_\gamma$ the corresponding column vector in bases $\beta$ and $\gamma$. We know we have the relations. $$X=P_\beta X_\beta=P_\gamma X_\gamma.$$

Now, in the standard basis, the linear transformation is represented by the relation $Y=LX$, which becomes $$P_\gamma Y_\gamma=LP_\beta X_\beta,\quad\text{whence }\quad Y_\gamma=P_\gamma^{-1} LP_\beta X_\beta, $$ so that the matrix $L_{\beta\gamma}$ of the linear transformation, with the input in basis $\beta$ and the output in basis $\gamma$ is simply $$L_{\beta\gamma}=P_\gamma^{-1} LP_\beta.$$