A bonus question given to us in Linear Algebra 2 course.
Let $A$ be any squared $n\times n$ matrix with real elements, then exists subspace $W \subset R$ of dimension 1 or 2 where $W$ is invariant under $A$.
First of all, I have no idea how to begin proving this, a hint is given.
Given a squared $n \times n $ matrix $A$ with real elements and vectors $a,b,c,d \subset R^n$ where $a+bi$ is eigenvector with eigenvalue $c+di$, show that $Span_R\{a,b\} \subset R^n$ is invariant under $A$ over field $R$.
I am stuck with both questions, I can't seem to prove either; I know it's somehow relevant to Jordan form and characteristic polynomial.
How would someone go on starting such question?