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Math Help - In need of someone to check my work.

  1. #1
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    In need of someone to check my work.

    So I just wanted to make sure I'm correct in my thinking for this problem.

    I'm trying to find a basis for the subspace W = { x in  R^{3} | x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, 2x + y = 0 }

    With this, we know x = -y /2 so therefore, a basis to the subspace =

     \begin{bmatrix} -1/2 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}


    is my work correct?
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  2. #2
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    Looks good.

    This subspace consists of all vectors of the form \left( \begin{array}{c} \frac{-1}{2}y\\y\\z\end{array}\right)=y\left( \begin{array}{c} \frac{-1}{2}\\1\\0\end{array}\right) +z\left( \begin{array}{c} 0\\0\\1\end{array}\right),

    so the vectors \left( \begin{array}{c} \frac{-1}{2}\\1\\0\end{array}\right) and \left( \begin{array}{c} 0\\0\\1\end{array}\right) span the subspace. Since they are obviously linearly independent, they form a basis for the subspace.
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  3. #3
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    Quote Originally Posted by topspin1617 View Post
    Looks good.

    This subspace consists of all vectors of the form \left( \begin{array}{c} \frac{-1}{2}y\\y\\z\end{array}\right)=y\left( \begin{array}{c} \frac{-1}{2}\\1\\0\end{array}\right) +z\left( \begin{array}{c} 0\\0\\1\end{array}\right),

    so the vectors \left( \begin{array}{c} \frac{-1}{2}\\1\\0\end{array}\right) and \left( \begin{array}{c} 0\\0\\1\end{array}\right) span the subspace. Since they are obviously linearly independent, they form a basis for the subspace.
    My thoughts exactly. Thanks a lot
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  4. #4
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    By the way, if you don't like fractions you could also say that y= -2x so that
    \begin{pmatrix}1 \\ -2 \\ 0\end{pmatrix} and \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix} are basis vectors.
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