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Math Help - Subspaces

  1. #1
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    Question Subspaces

    Hi I have two questions, I hope some one can help me.

    Which subsets of the vector space Mnn are subspaces?
    a) The set of all n x n symmetric matrices.
    b) The set of all n x n diagonal matrices.

    Thanks ^_^
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  2. #2
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    Once again all one must is to check two conditions:
    1) is the set closed with respect to addition?
    2) is the set closed with respect to scalar multiplication?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Once again all one must is to check two conditions:
    1) is the set closed with respect to addition?
    2) is the set closed with respect to scalar multiplication?
    Oh ok, can you give me a hint on how I can start it.
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  4. #4
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    You do know that if A is a symmetric matrix then A = A^T ?
    You should know that \left( {A + B} \right)^T  = A^T  + B^T \,\& \,\left( {\alpha A} \right)^T  = \alpha A^T.
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  5. #5
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    Quote Originally Posted by Plato View Post
    You do know that if A is a symmetric matrix then A = A^T ?
    You should know that \left( {A + B} \right)^T = A^T + B^T \,\& \,\left( {\alpha A} \right)^T = \alpha A^T.
    Yeah I know that the transpose of A = A.
    But How I can imply that to the subsets
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  6. #6
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    all A where A = A^T is the definition of your first subset. From there you need to prove that the summation of two elements from the subset results in an element of your subset and that any scalar multiplication also has the same result (definition of a subspace).

    e.g. Can you prove that (A+B) = (A+B)^T and \alpha \cdot A = \alpha \cdot A^T where A, B \in S_1; \alpha \in \Re ?
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  7. #7
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    Quote Originally Posted by malweth View Post
    all A where A = A^T is the definition of your first subset. From there you need to prove that the summation of two elements from the subset results in an element of your subset and that any scalar multiplication also has the same result (definition of a subspace).

    e.g. Can you prove that (A+B) = (A+B)^T and \alpha \cdot A = \alpha \cdot A^T where A, B \in S_1; \alpha \in \Re ?
    I got it.
    Thanks
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