Let $L:P_2(\mathbb{R})\rightarrow P_2(\mathbb{R})$ be a linear operator and $P_2(\mathbb{R})$ the real vectorspace containing real polynomials of degree < 2, where $L(\alpha + \beta X)=(3\alpha+2\beta)+(\alpha+2\beta)X$, for $\alpha,\beta\in\mathbb{R}$.
Find the eigenvalues for L and bases for the corresponding eigenspaces.
So I have already found the matrixrepresentation to be $_V[L]_V=\left[\begin{array}{l}3&2\\1&2\end{array}\right]$, where $V=(1,X)$ is a basis for $P_2(\mathbb{R})$. This means that characteristic polynomial is $t^2-5t+4$ and the eigenvalues are $1$ and $4$.
However I am not sure how to continue with the second part of the question. I have found eigenspaces
$E_{_V[L]_V}(1)=Span(\left[\begin{array}{l}-1\\1\end{array}\right])$ and
$E_{_V[L]_V}(4)=Span(\left[\begin{array}{l}2\\1\end{array}\right])$, But I still have to find $E_L(1)$ and $E_L(4)$.
Any help is appreciated!
We just require for example $L(\alpha+ \beta X) = 1(3\alpha+2\beta)+(\alpha+2\beta)X$.
(above is the eigenvector with eigenvalue $1$).
Solve to get $\alpha = 3\alpha+2\beta$, i.e. $\alpha = -\beta$,
and $\beta = \alpha+2\beta$, which results in the same relation.
Now our eigenvector is any scalar multiple of the polynomial $1-X$. That means that $1-X$ is a basis for the eigenspace with eigenvalue $1$.