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Vector spaces and spanning sets

  1. Mar 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Knowing this set spans M22:

    [1 , 0] , [0 , 1] , [0 , 0] ,[0 , 0]
    [0 , 0] , [0 , 0] , [1 , 0] ,[0 , 1]

    What is another spanning set for this vector space? Justify your choice by showing that it is a linearly independent set.

    3. The attempt at a solution

    [2 , 0] , [0 , 2] , [0 , 0] ,[0 , 0]
    [0 , 0] , [0 , 0] , [2 , 0] ,[0 , 2]

    I am I on the right track?
     
  2. jcsd
  3. Mar 16, 2009 #2

    Mark44

    Staff: Mentor

    This question is not well worded, IMO. You could add any old 2x2 matrix to the original set and still have a spanning set, but the new set would not be linearly independent, which is not a requirement of a spanning set. This is, however, a requirement of a minimal spanning set, which is the same as a basis.

    Your set of matrices is also a spanning set. To convince yourself of this show that any matrix M in M22 can be written as a linear combination of the elements in your set. I.e., c1*M1 + c2*M2 + c3*M3 + c4*M4 = M, where the Mi's are the matrices in your set.

    To show that your set of matrices is linearly independent, show that the equation
    c1*M1 + c2*M2 + c3*M3 + c4*M4 = 0 has exactly one solution: c1 = c2 = c3 = c4 = 0.
     
  4. Mar 16, 2009 #3
    Thanks for the help!
     
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