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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.

  1. The subset $\{A^2, A^5, A^{11}\}$ is always linearly dependent.
  2. The subset $\{I, A, A^2\}$ is always linearly dependent.
  3. It is possible that the subset $\{A^2, A^5, A^{11}\}$ is linearly independent.
  4. It is possible that the subset $\{I, A^2, A^5, A^{11}\}$ is linearly independent.
  5. 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!

  • 2
    I suppose you meant "linearly (in)dependent in the vector space of all $\,2\times 2\,$ real matrices", right? What have you done? Do you know something about a matrix characteristic polynomial?2012-10-11
  • 0
    What do you think is the dimension of the space of $2 \times 2$ matrices?2012-10-11
  • 1
    @wj32 That is not really apposite, because the [Cayley-Hamilton theorem](http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem) says that powers of an $n\times n$ matrix $A$ generate an $n$-dimensional subspace of the space of all $n\times n$ matrices.2012-10-11
  • 0
    @MJD: Oops, my bad!2012-10-11

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