<s>[INST] what is the image of the set $ \ A=\{(x,y)| x^2+y^2 \leq 1 \} \ $ under the linear transformation what is the image of the set $ \ A=\{(x,y)| x^2+y^2 \leq 1 \} \ $ under the linear transformation  $ \ T=\begin{pmatrix}1 & -1 \\ 1 & 1 \end{pmatrix} \ $
Answer:
From the given matrix , we can write as 
$ T(x,y)=(x-y,x+y)  \ $ 
But what would be the image  region or set $ \ T(A) \ $ ? 
 [/INST] \begin{align}T(x,y) &= (x-y, x+y)\\
&=\left(\begin{bmatrix} 1  & -1 \\ 1 & 1\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \right)^T \\
&=\sqrt{2} \left(\begin{bmatrix} \cos \left( \frac{\pi}{4}\right)  & -\sin \left( \frac{\pi}{4}\right) \\ \sin \left( \frac{\pi}{4}\right) & \cos \left( \frac{\pi}{4}\right)\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \right)^T\end{align}
Hence, the image will be $ T(A)=\{(x,y)| x^2+y^2 \leq 2 \} \ $
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