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Subspace or not?

  1. Oct 31, 2008 #1
    1. The problem statement, all variables and given/known data

    S is a subset of vector space V.

    If V = 2x2 matrix and S ={A | A is invertible}

    a) is S closed under addition?
    b) is S closed under scalar multiplication?

    2. Relevant equations

    3. The attempt at a solution

    For non singular 2x2 matrices, S is not closed under addition. but I am not quite sure about invertible 2x2 matrix.

    Say, A = [1 0]
    [0 1]

    So, if we add A + A, it is still invertible, so it is closed under addition. But does my statement lose the generality?
  2. jcsd
  3. Oct 31, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Your statement does lose the generality of the argument. Similarly I could argue if A is

    [1 0]
    [0 0]

    and B is

    [0 1]
    [0 0]

    A+B is non-invertible, hence non-singular matrices are closed under addition. But you know that's false. To prove the set of invertible matrices is closed under addition, you need to prove that given ANY A and B (not just a single example) A + B is invertible. Alternatively, you can find a single counterexample to prove that it is not closed under + (since if there exists A,B such that A+B is not invertible, then it is not true that A+B is invertible for all A,B)
  4. Oct 31, 2008 #3


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    Science Advisor
    Homework Helper

    What's the difference between 'non-singular' and 'invertible'? Aren't they the same thing?
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