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?


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.


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.