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Linear algebra-Basis of a linear map

  1. Jul 8, 2013 #1
    1. The problem statement, all variables and given/known data

    Let ##L: R^{2} → R^{2}## be a linear map such that ##L ≠ O## but## L^{2} = L \circ L = O.##
    Show that there exists a basis {##A##, ##B##} of ##R^{2}## such that:

    ##L(A) = B## and ##L(B) = O.##​
    3. The attempt at a solution
    Here's the solution my book provides :
    problem.JPG
    Well I have two questions:
    1.Why do they say that ##aA+bB=O##???. I mean I don't understand the solution from that point until the end (Why the solutions ##a=0## and ##b=0## are enough to prove the existence of that basis??May someone please explain??
    Thanks in advance :smile:. Any help would be appreciated
     
  2. jcsd
  3. Jul 8, 2013 #2
    The solution says IF [itex]aA+bB=0[/itex], THEN [itex]a=b=0[/itex]. That is what it means for the vectors [itex]A[/itex] and [itex]B[/itex] to be linearly independent. Vectors in a basis must be linearly independent.
     
  4. Jul 8, 2013 #3
    Thanks but why ##O=L(aA+bB)=aL(A)##?? could you please explain what they do there, please??
     
  5. Jul 9, 2013 #4
    [itex]L[/itex] is a linear map, which means [itex]L(aA+bB) = aL(A) + bL(B)[/itex].
     
  6. Jul 9, 2013 #5
    Thanks :smile:
     
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