Let C be an n×n matrix.Let W be the vector space spanned by {1,C,C^2,...,C^2n}.The dimension of W is?
1) 2n. 2) atmost n. 3)n^2. 4) atmost 2n
Plz help me...Thanks in advance for your time!
First, convince yourself of the following: Let $k$ be the smallest positive integer such that $\{I,\,C^1,\cdots, C^k\}$ is a linearly dependent set. Then the dimension of the space $W$ is $k$. (Part of this requires showing there is such a positive integer.)
Then the Cayley-Hamilton theorem says $k$ is at most $n$.