What is the transformation matrix $M$ that transforms a square in the $xy$-plane defined by $(1,1)^T,(-1,1)^T,(-1,-1)^T$ and $(1,-1)^T$ to a parallelogram whose corresponding vertices are $(2,1)^T,(0,1)^T,(-2,-1)^T$ and $(0,-1)^T$?

We need a transformation that maps each original point to the corresponding new point.
Check the coordinates:
$(1,1)\to(2,1)$
$(-1,1)\to(0,1)$
$(-1,-1)\to(-2,-1)$
$(1,-1)\to(0,-1)$
From these mappings, we can observe that $y'=y$.
Also, $x'=x+y$.
So, in homogeneous coordinate form:
$\left[\begin{matrix}x'\\y'\\1\end{matrix}\right]=\left[\begin{matrix}1&1&0\\0&1&0\\0&0&1\end{matrix}\right]\left[\begin{matrix}x\\y\\1\end{matrix}\right]$
Now verify using $(1,1)$:
$x'=1+1=2,\ y'=1$
So, $(1,1)\to(2,1)$.
Therefore, the required transformation matrix is:
$M=\left[\begin{matrix}1&1&0\\0&1&0\\0&0&1\end{matrix}\right]$