Studying for my final in linear algebra and came across this:

Problem: Let V be the vector space of all real n x n matrices over R. Let A be an element of V, and let W be the subspace of V spanned by $\displaystyle I, A, A^2, A^3 ...$. Prove that $\displaystyle dim(W) \leq n$.

Thoughts: The statement seems intuitive, but I'm not sure exactly what theorem/properties it results from. I know that a subspace cannot have dimension larger than its finite parent space, so $\displaystyle dim(W) \leq dim(V) = n$. However, my professor said that I should try to prove it by having it follow from the dimension of a matrix after raising it to higher powers; i.e. not from the obvious theorem.