$$T(x,y)=(2x+y+5,3x+2y+2)$$ Check if $T$ is affine and compute $T^{-1}$.
I'm not sure if I'm using the right methods here or if I am missing something, but I went about proving that $T(A)+T(B)=T(A+B)$ where $A=(x_1,y_1)$ and $B=(x_2, y_2)$. I then went on to show $c[T(A)]=T(cA)$ which I believe shows that $T$ is affine. Not too sure if this is the correct way to go about it and not sure where to go form here to get $T^{-1}$? Do I multiply by a $2×2$ matrix $A$ and leave it equal to the identity matrix?
Any assistance would be greatly appreciated!
$$T\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}2x+y+5\\ 3x+2y+2\end{pmatrix}=\begin{pmatrix}2x+y\\ 3x+2y\end{pmatrix}+\begin{pmatrix}5\\ 2\end{pmatrix}=\begin{pmatrix}2&1\\ 3&2\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}5\\ 2\end{pmatrix}$$ showing that $T$ is affine. To calculate the inverse, we have to invert $$A=\begin{pmatrix}2 & 1\\ 3&2\end{pmatrix}$$ first. $$A^{-1}=\begin{pmatrix}2 & -1 \\-3& 2\end{pmatrix}$$ and we get the inverse affine transformation $T^{-1}$ by $$T(x,y)=A^{-1}\left(\begin{pmatrix}x\\ y\end{pmatrix}-\begin{pmatrix}5\\ 2\end{pmatrix}\right)=A^{-1}\begin{pmatrix}x\\ y\end{pmatrix}-A^{-1}\begin{pmatrix}5\\2\end{pmatrix}\\=\begin{pmatrix}2 & -1\\ -3& 2\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}-\begin{pmatrix}8\\ -11\end{pmatrix}.$$