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Math Help - Help please!

  1. #1
    Junior Member
    Joined
    Jul 2006
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    73

    Help please!

    Let W be the subspace of R^4 spanned by :
    [ 2
    0
    -1
    3]
    and
    [ -6
    1
    5
    -8]
    Find a basis for W^(upside down T)

    dont know the correct word for the symbol that looks like an Upside down T
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by luckyc1423
    Let W be the subspace of R^4 spanned by :
    [ 2, 0, -1, 3]' and [-6, 1, 5, -8]'

    Find a basis for W^(upside down T)

    dont know the correct word for the symbol that looks like an Upside down T
    \{[1,\ 0,\ 0,\ 0]',\ [0,\ 1,\ 0,\ 0]', \ [ 2,\ 0,\ -1,\ 3]',\ [-6,\ 1,\ 5,\ -8]'\}=\{e_1,e_2,w_1,w_2\}

    is a basis of \mathbb{R}^4 (proof left as an excercise for the reader).

    Now w_1,\ w_2 is a basis for W, so [w_1,w'_2],
    where:

    <br />
w'_2=w_2-\frac{\langle w_1,w_2\rangle w_1}{\|w_1\|^2}<br />

    is an orthogonal basis for W.

    Now let:

    <br />
e'_1=e_1-\frac{\langle e_1,w_1\rangle w_1}{\|w_1\|^2}-\frac{\langle e_1,w'_2\rangle w'_2}{\|w'_2\|^2}<br />

    and:

    <br />
e'_2=e_2-\frac{\langle e_2,w_1\rangle w_1}{\|w_1\|^2}-\frac{\langle e_2,w'_2\rangle w'_2}{\|w'_2\|^2}}-\frac{\langle e_2,e'_1\rangle e'_1}{\|e'_1\|^2}<br />
.

    Now if I have done this right we should have \{e'_1,\ e'_2,\ w_1,\ w'_2\}
    is an orthogonal basis of \mathbb{R}^4, with \{w_1,\ w'_2\} an orthogonal basis
    of W, and \{e'_1,\ e'_2\} and orthogonal basis of W^{\bot}.

    RonL
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