Let $\displaystyle T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\displaystyle T(x,y)=(3x+y, 2x+2y)$.

Show that there exists a basis $\displaystyle \bold {B} =\{ \alpha _1 , \alpha _2 \}$ of $\displaystyle \mathbb{R}^2$ formed by eigenvectors.

Find the matrix of $\displaystyle T$ with respect of the canonical basis and the matrix of $\displaystyle T$ with respect of the basis of eigenvectors of $\displaystyle \bold {B}$.

My attempt: I don't know how to show the first part.

I think I've found the matrix of $\displaystyle T$ with respect to the canonical basis :

$\displaystyle T(1,0)=(3,2)$

$\displaystyle T(0,1)=(1,2)$. Hence the matrix is $\displaystyle \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix}$. Let's call it $\displaystyle T'$.

I don't know how to get it with respect to another basis, even if I realize I must know it since it's very basic.

From memory, I think that eigenvalues of $\displaystyle T$ are the $\displaystyle \lambda$ in $\displaystyle \det [I \lambda - T']=0 \Leftrightarrow \lambda=1 \text{ or } 4$. I guess that by finding the eigenvalues or $\displaystyle T$ it would help to find its eigenvectors... I'm lost! I'd appreciate a bit of help if you can. Thanks.