I have a linear operator $L:P_2(\mathbb{R})\to P_2(\mathbb{R})$ where
$L(\alpha+\beta X)=(3\alpha+2\beta)+(\alpha+2\beta)X, \alpha ,\beta\in\mathbb{R}$,
I want to find the following for $L$:
The characteristic polynomial for $L$
Eigenvalues for $L$ and bases for the eigenspaces.
In order to find the characteristic polynomial, I first need to find the standard matrix representation (SMR) of $L$. A basis for $P_2(\mathbb{R})$ is $\mathcal{V}=(1,X)$. That gives me an isomorphism:
$L_{\mathcal{V}}:\mathbb{R}^2\to P_2(\mathbb{R})$.
How do I continue from here?
I really want to understand the process of finding the bases for the eigenspaces.
Observe that $L(1) = 3+X$ and $L(X) = 2+2X.$ So the standard matrix representation of $L$ is $$A=\begin{pmatrix} 3 & 2 \\ 1 & 2 \\ \end{pmatrix}.$$
So the characteristic polynomial of $L$ is $x^2 - 5x+ 4=0$ and hence the eigenvalues are $1$ and $4.$
Now consider the equations $$\begin{pmatrix} 3 & 2 \\ 1 & 2 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} x \\ y \\ \end{pmatrix}.$$ and $$\begin{pmatrix} 3 & 2 \\ 1 & 2 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = 4 \begin{pmatrix} x \\ y \\ \end{pmatrix}.$$ Find any non-zero solutions to both of the equations to get the bases for the eigenspace of $L.$
You may check that $\left \{\begin{pmatrix} 1 \\ -1 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} \right \}$ is one of the bases for the eigenspace of $L.$