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Math Help - basis for matrix

  1. #1
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    basis for matrix

    I have this matrix made of two spanning sets:
    1 1 3 2 0 8
    5 4 14 3 1 9
    -2 0 -4 -2 0 -8
    3 3 9 0 1 3

    In reduced row echelon form the matrix is:
    1 0 2 0 1/6 0
    0 1 1 0 1/6 0
    0 0 0 1 -1/6 0
    0 0 0 0 0 1

    W1 is the span of the first three columns of the original matrix.
    W2 is the span of the last three columns of the original matrix.

    I need to find a basis for:
    W1
    W2
    W1+W2
    the intersection of W1 and W2

    I think I'm wrong but I said that:
    a basis for W1= {Column 1, Column 2}
    a basis for W2={Column 4, Column 5}
    a basis for W1+W2={Column 1, Column 2, Column 4, Column 6}
    this means that there would be no basis for the intersection of W1 and W2

    doesn't make sense
    could someone check this?
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  2. #2
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    Could anyone just see if this makes sense?
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  3. #3
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    Quote Originally Posted by PvtBillPilgrim View Post
    I have this matrix made of two spanning sets:
    1 1 3 2 0 8
    5 4 14 3 1 9
    -2 0 -4 -2 0 -8
    3 3 9 0 1 3

    In reduced row echelon form the matrix is:
    1 0 2 0 1/6 0
    0 1 1 0 1/6 0
    0 0 0 1 -1/6 0
    0 0 0 0 0 1

    W1 is the span of the first three columns of the original matrix.
    W2 is the span of the last three columns of the original matrix.

    I need to find a basis for:
    W1
    W2
    W1+W2
    the intersection of W1 and W2

    I think I'm wrong but I said that:
    a basis for W1= {Column 1, Column 2}
    a basis for W2={Column 4, Column 5}
    a basis for W1+W2={Column 1, Column 2, Column 4, Column 6}
    this means that there would be no basis for the intersection of W1 and W2

    doesn't make sense
    could someone check this?
    I don't have time to work it all out, but looks like you're right on W1 and W1+W2, except for W2 you need all three columns because those three are all linearly independent.

    From my initial glance, it looks like a basis for the intersection is the single vector <1,1,0,0> (not an actual column from ths matrix). This is because W1 has vectors looking like <a,b,0,0> and W2 has vectors looking like <c,c,d,e>.

    Sorry I don't have time to explain in more detail. I hope this helps.
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