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Determining if a vector b is a linear combination of vectors a1,a2,a3

  1. Sep 17, 2011 #1
    given a1=[1,-2.0] , a2=[0,1,2] , a3=[5,-6,8] , b=[2,-1,6] determine if b is a linear combination of a1, a2, a3.

    I put these vectors into an augmented matrix and row reduced. My result was columns 1 and 2 having pivots and the last row being all zeros. Please help me understand how this results in b NOT being a linear combination of a1, a2, a3.

    The only thing i can think of is that vectors a1, a2, a3 do not span all of R3 and b is not part of the plane that a1, a2, a3 span. If this is the case, please explain how i can see this from the augmented matrix in RREF.
     
    Last edited: Sep 17, 2011
  2. jcsd
  3. Sep 18, 2011 #2
    You've misinterpreted the results of the row reduction.

    First lets start with the result:
    [itex] \left[ \begin {array}{cccc} 1&0&5&2\\ -2&1&-6&-1
    \\ 0&2&8&6\end {array} \right]
    \to \left[ \begin {array}{cccc} 1&0&5&2\\ 0&1&4&3
    \\ 0&0&0&0\end {array} \right]
    [/itex]

    So the coefficients of a linear combination [itex]x_1a_1 + x_2a_2 + x_3a_3 = b[/itex] take the form:
    [itex]
    \left[\begin{array}{c}
    x_1 \\ x_2 \\ x_3
    \end{array}\right] \in
    \left\{
    \left[\begin{array}{c}
    2 \\ 3 \\ 0
    \end{array}\right] +
    t\left[\begin{array}{c}
    -5 \\ -4 \\ 1
    \end{array}\right] \|
    t \in \mathbb{F}
    \right\}
    [/itex]
     
  4. Sep 18, 2011 #3
    ok. that clears things up. thanks so much
     
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