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Math Help - [SOLVED] Show that the following vectors are two different bases for the following pl

  1. #1
    Newbie JCS007's Avatar
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    [SOLVED] Show that the following vectors are two different bases for the following pl

    Show that

    \begin{pmatrix} 4\\0\\1 \end{pmatrix} , \begin{pmatrix} 2\\1\\0 \end{pmatrix}
    and
    \begin{pmatrix} 2\\-1\\1 \end{pmatrix} , \begin{pmatrix} 0\\2\\-1 \end{pmatrix}

    are two different bases for the plane

    x-2y-4z=0

    Thanks in advance!
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JCS007 View Post
    Show that

    (4 0 1) .^T , (2 1 0) .^T and
    (2 -1 1) .^T , (0 2 -1) .^T

    are two different bases for the plane

    x-2y-4z=0

    Thanks in advance!
    Consider an arbitrary vector written in terms of the first basis:
    V = a \left ( \begin{matrix} 4 \\ 0 \\ 1 \end{matrix} \right ) + b \left ( \begin{matrix} 2 \\ 1 \\ 0 \end{matrix} \right )

    Now show that V_x - 2V_y - 4V_z = 0 for all a and b.

    The proof for the other basis is similar.

    -Dan
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