Let $C\in \text{Mat}_{n\times n}(\mathbb R)$. Then which of the alternatives are correct:
- $\operatorname{dim}\langle I,C,C^2,\dots,C^{2n}\rangle$ is at most $2n$
- $\operatorname{dim} \langle I,C,C^2,\dots,C^{2n}\rangle$ is at most $n$.
Let $C\in \text{Mat}_{n\times n}(\mathbb R)$. Then which of the alternatives are correct:
Following Julian's suggestion:
Cayley Hamilton shows that every matrix satisfies its own characteristic polynomial, in particular, $p(C)=0$ for some polynomial $p$ of degree $n$. Hence $C^n$ can be written as a linear combination of $I,C,...,C^{nā1}$.
It follows by induction that any higher power of $C$ can be written in terms of $I,C,...,C^{nā1}$.
Hence both statements are correct, as Marc pointed out, but the second is stronger.