$\begin{bmatrix} -2 & 11 \\ 4 & 2 \end{bmatrix}$ represents a linear transformation $T:$$\mathbb{R}^2$ to $\mathbb{R}^2$ with respect to the basis, ${[(3,1),(0,2)]}$. Find the matrix of $T$ with respect to basis ${[(1,1),(-1,1)]}$
There is some problem with the approach I'm employing, I'm not sure what. I am a little confused with these tyeles of questions are to be approached. Please help!
Best Answer
The change of basis matrix from $\bigl((1,1),(-1,1)\bigr)$ to $\bigl((3,1),(0,2)\bigr)$ is$$P=\begin{bmatrix}\frac13&\frac13\\\frac13&-\frac23\end{bmatrix}.$$So, the answer to your problem is the matrix$$P^{-1}\begin{bmatrix}-2&11\\4&2\end{bmatrix}P=\begin{bmatrix}8&-16\\1&-8\end{bmatrix}.$$