Dimension of space composed of powers of a matrix

[Grothard]

Let A be an n*n matrix.
Consider the space span \{ I, A, A^2, A^3, ... \} .
How would one show that the dimension of the space never exceeds n?
I feel like the answer lies somewhere near the Cayley-Hamilton theorem, but I can't quite grasp it.

 

[micromass]

The answer is indeed given by Cayley-Hamilton.

Basically, what you must prove is that every matrix like Aⁿ, Aⁿ⁺¹, etc. can be written as a linear combination of the matrices I,A,...,A^n-1.

Now, what does Cayley-Hamilton say?? Doesn't this theorem answer our question?

How would you use Cayley-Hamilton to write Aⁿ as a linear combination of I,A,...,A^n-1?