The following questions are not T/F questions.
I'm trying to understand this complex subsets independency or dependency.
Let $A$ be an $2\times 2$ matrix over the real numbers.
- The subset $\{A^2, A^5, A^{11}\}$ is always linearly dependent.
- The subset $\{I, A, A^2\}$ is always linearly dependent.
- It is possible that the subset $\{A^2, A^5, A^{11}\}$ is linearly independent.
- It is possible that the subset $\{I, A^2, A^5, A^{11}\}$ is linearly independent.
- It is possible that the subset $\{I, A, A^2\}$ is linearly independent.
$I$ refers to the identity matrix.
Examples would be appricated.
Thanks in advance!