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    Linear algebra: Dimensions

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    Find the dimensions of each of the following vector spaces.

    a) The vector space of all diagonal n X n matrices

    b) The vector space of all symmetric n X n matrices

    c) The vector space of all upper triangular n X n matrices

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    Solution Preview

    The dimension of a vector space is the number of the basis vectors.
    For example, let's look at the general space (all the matrices).
    They are of the form:
    Since we have four independent variables (the matrix entry) we have four degrees of freedom and therefore we have four basis vectors.
    In this case, the standard basis is:
    It is easy to see that indeed since:
    And the basis matrices are linearly independent since if

    We must have
    Of course we can come up with other basis that will span this space, but we know that all of the bases must have the same number ...

    Solution Summary

    The expert examines linear algebra for dimensions. Vector space is examined.