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Linear Algebra question: finding bases

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data

    The problem states:


    A_1 = [-1 1] , A_2 = [1 3]
    .........[0 1]............[-1 0]

    A_3 = [1 0] , A_4 = [0 -1]
    .........[1 2]............[2 3]

    Show that {A_1, A_2, A_3, A_4} is a basis for M_2 R.

    The attempt at a solution

    I'm very confused about this problem. I understand that to show {A_1, ..., A_4} is a basis, I must show 1.) the set is linearly independent, and 2.) it is a spanning set; however, I know there is a less complicated way instead of going through these 2 steps. I'm really not sure what the "easy" way is for doing this problem... it hints that there is a Thm that will help solve the problem, but I have found none that fit the bill. Can someone please help me with all this?
  2. jcsd
  3. Oct 16, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    A basis, for a vector space of dimension n, satisfies 3 properties:
    1) it spans the space
    2) it is independent
    3) it contains n vectors.

    If any two of those are true the third must be. If you know, or are given, that M_2 R is a 4 dimensional vector space, that might be the theorem you are referring to. Since this set contains 4 matrices, (3) above is clearly satisfied and you only need to prove either (1) or (2), not both.

    If you do not know that the M_2 R is 4 dimensional, then you will need to prove both (1) and (2), and, honestly, they are not that difficult! Both reduce to 4 equations in 4 unknowns and you really only need to show that those equations have a unique solution- which is true if the 4 by 4 coefficient matrix does not have 0 determinant. It would probably take you less time than waiting for a response on the internet.
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