Let C be amatrix. Let W be the Vector space spanned by {I, C, Cn x n^{2}, C^{3}, .... C^{2n}}. What is the dimension of the vector space W?

My argument is as follows.

Argument- By Cayley hamilton theorem, we have a degree of equation 'n' satisfied by the matrix C.

This is of the form a_{n}C^{n}+ a_{n-1}C^{n-1}+ a_{n-2}C^{n-2}+ +..... a_{0}I = 0. which implies the n+1 elements, namely, I, C, C^{2}, C^{3},...C^{n}are linearly dependent or (one-less) no. of elements is linearly independent.

Therefore, n+1-1 = n is the basis of the vector space.

Are there any other possible explanations? answers?