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Math Help - Finding Basis'

  1. #1
    Junior Member
    Joined
    Aug 2010
    Posts
    31

    Finding Basis'

    "Working over the real numbers
    R, let U denote the solution set of the system of equations
    3
    x1 + x2 + 7x3 + 4x4 = 0
    5
    x1 + 6x2 + 7x3 + 3x4 = 0
    and let
    W denote the solution set to the system of equations
    6
    x1 + 4x2 + 3x3 + 6x4 = 0
    4
    x1 x2 + 3x3 + 7x4 = 0.

    Construct bases for
    U, W, U W and U +W.

    "

    Im think i'm doing this right, but not 100% sure.

    To find basis for U:
    Write solution set in augmented coefficent form
    Get in row reduced echleon form
    Write two of (x1, x2, x3, x4) in terms of other two. Then re-write in
    (x1, x2, x3, x4) form.
    Then seperate the two terms like(not from example) "x2( 3, 0, 1, 11) + x3( 4, 7, 0 9)".
    Which givesthe basis (3, 0, 1, 11) & (4, 7, 0 9).

    Do the same for W

    For U+ W, add basis together then apply a pruning process.

    For U intersection W, I don't have a clue!

    Would be very grateful for any help

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  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
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    45
    We have

    U\cap V\equiv \begin{Bmatrix}3x_1+x_2+7x_3+4x_4=0\\ 5x_1+6x_2+7x_3+3x_4=0\\6x_1+4x_2+3x_3+6x_4=0\\4x_1-x_2+3x_3+7x_4=0\end{matrix}

    Now, transform the system into row echelon form etc.
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