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Is S a Subset of R^(2x2)?

  1. Feb 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose A is a vector [tex]\in[/tex] [tex]R^{2x2}[/tex].

    Find whether the following set is a subspace of [tex]R^{2x2}[/tex].

    [tex]S_{1} = {B \in R^{2x2} | AB = BA}[/tex]


    3. The attempt at a solution
    I know that S isn't empty, because the 2 x 2 Identity matrix is contained in S.

    The problem I'm having comes in the proof that addition is closed.

    If I show A(B + C) = (B + C)A that should be sufficient, right?

    So far I have:

    Suppose [tex]B[/tex] and [tex]C[/tex] [tex]\in[/tex] S.
    [tex]A(B + C) = (B + C)A[/tex]
    [tex]AB + AC = BA + CA[/tex]

    And that's where I'm stuck. I have no idea where to continue on to. Any help would be greatly appreciated.
     
  2. jcsd
  3. Feb 18, 2010 #2

    Mark44

    Staff: Mentor

    You need to show that S is closed under addition and scalar multiplication. You probably want to do those separately.
     
  4. Feb 18, 2010 #3
    I know that's what I have to do, but I don't know how to go about doing it. I started the addition part up above, and am stuck at that point.
     
  5. Feb 19, 2010 #4

    Mark44

    Staff: Mentor

    OK, B and C are both elements of S.
    A(B + C) = AB + AC (since vector multiplication is left-distributive)
    Now, what can you say about AB and AC, since B and C are members of set S?
     
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