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Is the methodology to change the basis of a matrix the same as changing the basis of a vector? For example, if I had $A : \mathbb{R}^2 \to \mathbb{R}^2$ $$A=\begin{pmatrix} 3 & -5 \\ 2 & 7 \end{pmatrix}$$ in the standard basis and wanted it in the basis $v_1 = (1,3), v_2=(2,5)$. To do this, I simply multiply $A * \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}$ to get $\begin{pmatrix} -12 & -19 \\ 23 & 39 \end{pmatrix}$? Is this correct?

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Almost there, if you have a matrix $A$ with respect to standard basis and $D$ is the matrix of the transformation with respect to the basis, say $B=\lbrace v_1,v_2\rbrace \subset \mathbb{R}^2$ (Notice that $v_1$ and $v_2$ are L.I) then, after finding $C = \begin{pmatrix}1&2\\3&5\end{pmatrix}$ the change of basis matrix for basis B, you want to find $D$ in terms of $C$ and $A$ as follows

$$D\ [\vec{x}]_B = [T\ \vec{x} ]_B = [A\ \vec{x}]_B = C^{-1} A\ [\vec{x}] = C^{-1} A\ C\ [\vec{x}]_B$$

where $[\vec{x}] = C\ [\vec{x}]_B$ and $T \ \vec{x} = A \ \vec{x}$ the last one in standard coordenates.

Check this video to get more conclusions.

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