how to Find all T-invariant subspaces of T?

Question

Let T be the linear transformation on R2 which is represented by the matrix

$$ \begin{matrix} 1 & 4 \\ 2 & 3 \\ \\ \end{matrix} $$ Find all T-invariant subspaces.

I understand that span $ \begin{matrix} 1 \\ 1 \\ \\ \end{matrix} $ is one of them- how do I find the rest?

Answer 1

Tht matrix has two eigenvalues: $5$ and $-1$. An eigenvector corresponding to the eigenvalue $5$ is the one that you mentioned: $(1,1)$. An eigenvector corresponding to the eigenvalue $-1$ is $(2,-1)$. Therefore, the invariant subspaces are:

• $\{0\}$;
• $\mathbb R(1,1)$;
• $\mathbb R(2,-1)$;
• $\mathbb R^2$.