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Vector spaces

  1. Apr 19, 2007 #1

    daniel_i_l

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    1. The problem statement, all variables and given/known data
    a)A is an invertible 5x5 matrix with complex elements. Find all the values of det(A) where adj(adj(A)) = A. Use the definition of the adjoint (transpose of cofactor matrix)

    b) u1,u2,u3 are linearly independent vectors in V. We define:
    v1=u1+u2+u3, v2=u1-u2+u3, v3=-u1+3u3-u3
    Does Sp{u1,u2,u3}=Sp{v1,v2.v3} ?


    2. Relevant equations
    a) The definition of adj(A)
    b) If A and B are subsets of space V then Sp(A) = SP(B) iff A is in Sp(B) and B is in Sp(A).


    3. The attempt at a solution
    a) I tried to use the definition of the adjoint to find the adjoint of adj(A) but it quickly got so complicated that I couldn't see how to calculate det(A) from it. Is there a way to use the fact that A is a 5x5 matrix to simplify things?
    b)I think that the answer is yes because obviously {v1,v2,v3} is in the span of {u1,u2,u3} because of their definition. And by solving the equation:
    [v1|v2|v3][x]=[un] for all 0<n<=3 I can show that every u is a linear combination of v's and so {u1,u2,u3} is in Sp{v1,v2,v3}. Is that right?
    Thanks.
     
  2. jcsd
  3. Apr 19, 2007 #2

    matt grime

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    When you say you used the definition of adj(A), do you mearn you used the definition, or the formula? adj(A), is the (unique for det=/=0) matrix satisfying A*adj(A)= det(A)*Id.
     
  4. Apr 19, 2007 #3

    daniel_i_l

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    I mean the formula: [tex] [adjA]_{ij} = (-1)^{i+j}*M_{ji} [/tex]
    Where [tex] M_{ji} [/tex] is the minor (ji). (by minor (j,i) I mean the det of the matrix obtained by taking away the jth row and the ith coloum)
     
  5. Apr 19, 2007 #4

    matt grime

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    yes, I know that what was you used. I didn't ask for my benefit.
     
  6. Apr 19, 2007 #5

    HallsofIvy

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    Now, what relation does det(A) have to [tex] [adjA]_{ij} = (-1)^{i+j}*M_{ji} [/tex]?

    As for B, you were told that u1, u2, and u3 are linearly independent. That means they are a basis for their span which must be of dimension 3. The real question is "Are v1, v2, and v3 linearly independent?"
     
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