Let A be a real n x n matrix. Prove that we can find a subspace V in R^N such that 1 <= dim V < = 2 and A(V) is a subset of V.
None I don't think.
The Attempt at a Solution
I know that the eigenspace of a matrix satisfies the condition that A(E) is a subset of E since for any vectors v in the eigenspace, Av = λv. Since we know v is in the eigenspace, any multiple of v by λ is also in the eigenspace. Now how am I supposed to incorporate the dimension into the argument?