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Math Help - Basis, set of matrices, proof...

  1. #1
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    Basis, set of matrices, proof...

    If I have a set of 2X2 matrices A1, A2, A3, A4 and I want to show that they form a basis for the set of all 2X2 matrices M_{22}, do I need to show that the set spans M_{22}, and is linearly independent? Or can I just show that they are linearly independent to prove that they are a basis for M_{22}??? Don't I know that M_{22} is a four dimensional vector space, and since I have a set of 4 matrices, I only need to show spanning or linear independence right?

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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by adkinsjr View Post
    ... M_{22} is a four dimensional vector space, and since I have a set of 4 matrices, I only need to show spanning or linear independence right?

    Right.
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  3. #3
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    A basis for any n dimensional vector space (finite dimensional, of course) has 3 properties
    1) They span the space
    2) They are independent
    3) There are n vectors in the basis

    And if any two of those are true, the third is also.
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