Q:

Linear Algebra Multiple Choice question. $A$ be a $5\times 5$ matrix, consider $\{x,Ax,..A^5x\}$

$A$ be a $5\times 5$ matrix with real entries. Consider the set $\{x,Ax, A^2x, A^3x, A^4x,A^5x\}$. What can we say about the set? Is it linearly independent or linearly dependent? When it would be Linear Independent?
My Efforts
First of all assume on the contrary that this set was linearly independent then It would contradict the fact that $\mathbb{R}^5$ has dimension $5$.
So this is set is linearly dependent. 
Further Analysis
Now I have some question?
Is the set $\{x, A^2x, ..,A^4x\}$ linearly independent? I know it is not true for in general matrix as, if you take identity matrix, you get a contradiction. 
So under what condition on $A$ this set become linearly independent?
Suppose $A$ is a general $n\times n$ matrix. Then I want to say something similar about $\{x,A^2x,...A^{n-1}x\}$.

A:

What you are considering is called a cyclic subspace, and you want to know when the matrix $A$ has a cyclic vector, i.e., such that its cyclic subspace generates the whole vector space.
A matrix has a cyclic vector iff it is similar to the companion matrix of a polynomial; in general, a matrix is similar to a matrix formed by blocks by companion matrices (see rational canonical form), so you are asking for it to have only one such block.

