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Math Help - Linear algebra plane span

  1. #1
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    Linear algebra plane span

    Let  a_1=  \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}, a_2=  \begin{bmatrix} -2 \\ -3 \\ 7 \end{bmatrix} and  b= \begin{bmatrix} 4 \\ 1 \\ h \end{bmatrix} . For what value(s) of h is b in the plane spanned by  a_1 and a_2?
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  2. #2
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    Quote Originally Posted by Linnus View Post
    Let  a_1=  \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}, a_2=  \begin{bmatrix} -2 \\ -3 \\ 7 \end{bmatrix} and  b= \begin{bmatrix} 4 \\ 1 \\ h \end{bmatrix} . For what value(s) of h is b in the plane spanned by  a_1 and a_2?
    You are asking for what h does there exists real numbers a,b so that a\bold{a_1}+b\bold{a_2} = \bold{b}?

    This leads to the equations,
    a-2b = 4
    4a -3b = 1
    -2a+7b = h

    We want this system to be consistent i.e. it has solutions.
    Convert this equation into a matrix,
    \begin{bmatrix}1 & -2 & 4 \\ 4 & - 3 & 1 \\ -2 & 7 & h \end{bmatrix}

    Now use Gauss-Jordan elimination to reduce it you should get on the bottom row something like: [ 0 ~ \ ~ 0 ~ \ ~ \text{ expression of h} ].
    Then in order for it to be consider that expression of h needs to be equal to zero.
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