I got this problem and the answer the probably something with fundamental definition. Please help me out. I am a beginner in Linear Algebra.
So here is my problem:
Let A be an $nxn$ matrix. Prove that
$dim(span({I_n,A,A^2,...}))≤n$
I was thinking that we could use the definition of span, which is a subspace of all vectors that can be linearly represented by ${I_n,A,A^2,...}$, but I do not know how to move on.
I also thought it probably has something to do with charactericstic polynomial (since this exercise comes from this chapter), but I have no clue of any possible connection.
Thank you for your help!
According to Cauchy-Hamilton's theorem about matrices, the powers of $A$ after $A^n$ can be expressed in terms of powers of $A$ from $0$ to $n-1$. Then what you need to prove, is $$ \dim\{I_n,A,A^2,\cdots,A^{n-1}\}\le n. $$