# Finding Matrix of T with respect to basis.

$$\begin{bmatrix} -2 & 11 \\ 4 & 2 \end{bmatrix}$$ 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!

## 1 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$$P^{-1}\begin{bmatrix}-2&11\\4&2\end{bmatrix}P=\begin{bmatrix}8&-16\\1&-8\end{bmatrix}.$$

• Shouldn't the basis change from the second to the first? I'm sorry, my concepts are a bit shaky. – Shinjini Rana Nov 8 '18 at 12:04
• I understand now. That was a stupid doubt. Thanks for the answer! – Shinjini Rana Nov 8 '18 at 12:07
• I'm glad I could help. – José Carlos Santos Nov 8 '18 at 12:08