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Thread: Help please!

  1. #1
    Junior Member
    Joined
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    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
    $\displaystyle \{[1,\ 0,\ 0,\ 0]',\ [0,\ 1,\ 0,\ 0]',$$\displaystyle \ [ 2,\ 0,\ -1,\ 3]',\ [-6,\ 1,\ 5,\ -8]'\}=\{e_1,e_2,w_1,w_2\}$

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

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

    $\displaystyle
    w'_2=w_2-\frac{\langle w_1,w_2\rangle w_1}{\|w_1\|^2}
    $

    is an orthogonal basis for $\displaystyle W$.

    Now let:

    $\displaystyle
    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}
    $

    and:

    $\displaystyle
    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}
    $.

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

    RonL
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