Given the standard matrix of a linear mapping, determine the matrix of a linear mapping with respect to a basis

Question

If we let \begin{bmatrix}-1 & 3 \\-3 & 2 \\\end{bmatrix} be the standard matrix of a Linear Mapping L: R2 -> R2. How do we determine the matrix of L with respect to the basis B = \begin{Bmatrix}\begin{bmatrix}1 \\5\\\end{bmatrix},\begin{bmatrix}2 \\7\\\end{bmatrix}\\\end{Bmatrix}

Answer 1

Consider $ \begin{bmatrix}1& 5 \\ 2& 7\end{bmatrix} $.

This changes basis from the basis $\left \{\begin{bmatrix} 1\\2 \end{bmatrix},\begin{bmatrix} 5\\7 \end{bmatrix}\right\}$ to the standard basis...

$ \begin{bmatrix}1& 5 \\ 2& 7\end{bmatrix} ^{-1}=\begin{bmatrix} -\frac73 &\frac53\\\frac23 &-\frac13 \end{bmatrix}$

Now we want $ \begin{bmatrix}1& 5 \\ 2& 7\end{bmatrix} \begin{bmatrix}-1& 3 \\ -3& 2\end{bmatrix} \begin{bmatrix} -\frac73 &\frac53\\\frac23 &-\frac13 \end{bmatrix} $.

I get: $\begin{bmatrix} -16& 13 \\-23&20\end{bmatrix}\begin{bmatrix} -\frac73 &\frac53\\\frac23 &-\frac13 \end{bmatrix}=\begin{bmatrix}22& -31\\67&-45\end{bmatrix} $