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Thread: dimension of the vector space

  1. #1
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    dimension of the vector space

    Let C be an nn 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!
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  2. #2
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    Re: dimension of the vector space

    First, do you know what the dimension of the space of all n by n matrices?

    If you do consider the case when C is the identity matrix.
    Thanks from topsquark
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  3. #3
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    Re: dimension of the vector space

    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 $n$.
    Thanks from snehal
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