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Math Help - constructing basis

  1. #1
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    constructing basis

    Define a real linear transformation
    L1 : R4 \to R2 by

    L
    1(x1, x2, x3, x4) = (3x1 + x2 + 2x3 x4, 2x1 + 4x2 + 5x3 x4)

    and let
    U1 denote the kernel of L1. Define a real linear transformation L2 : R4 ! R2 by

    L
    2(x1, x2, x3, x4) = (5x1 + 7x2 + 11x3 + 3x4, 2x1 + 6x2 + 9x3 + 4x4)

    and let
    U2 denote the kernel of L2.

    Construct bases for U1, U2, U1 &U2 and U1 + U2.


    Not sure on how to tackle this question. Or how to construct basis in similar questions. Any help would be appreciated a lot!
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  2. #2
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    The kernel of a linear transformation, L, is, by definition, the space of vectors, v, such that L(v)= 0. If (x_1, x_2, x_3, x_4) is in the kernel of L_1 then 3x_1 + x_2 + 2x_3- x_4= 0, 2x_1 + 4x_2 + 5x_3- x_4= 0.

    We can solve those two equations for two of the unknowns in terms of the other two. For example If we subtract the second equation from the first, we eliminate x_4: x_1- 3x_2- 3x_3= 0 so that x_1= 3x_2+ 3x_3. Putting that back into the second equation, [tex]2(3x_2+ 3x_3)+ 4x_2+ 5x_3- x_4= 10x_2+ 11x_3- x_4[tex] so that x_4= 10x_2+ 11x_3.

    That is, we can write (x_1, x_2, x_3)= (3x_2+ 3x_3, x_2, x_3, 10x_2+ 11x_3)= (3x_2, x_2, 10x_2)+ (3x_3, 0, x_3, 11x_3)= x_2(3, 1, 0, 10)+ x_3(3, 0, 1, 11)

    Now, do you see what a basis of U_1 is?

    Do the same thing for U_2.

    To find a basis for U_1+ U_2 (I assume you mean the direct sum), combine the two bases, then throw out any that can be written as a linear combination of others. I don't know what " U_1& U_2" means.
    Last edited by HallsofIvy; May 10th 2011 at 02:53 PM.
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  3. #3
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    by U1&U2, he probably means the intersection.

    by U1+U2, he probably means the sum (not the direct sum).

    these would be the meet and join of the subspaces in the lattice of subspaces of R^4.
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  4. #4
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    Thanks that helped.

    So the basis of U_1 is : {(3,1,0,10),(3,0,1,11)}

    And yes sorry i meant the intersection.

    So the basis of U_1 + U_2 , is the combined basis, but then I should apply the pruning process to get rid of some?

    How do i get the basis of intersection of U_1 U_2 ?
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  5. #5
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    Im having problems finding the basis from U2. Is there not a process i should be using like matrices, rather than just doing it by inspection.
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