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Math Help - Spans

  1. #1
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    Spans

    Find all values a for which \begin{bmatrix}1  \\ 1 \\ 1\end{bmatrix} is in the span of v_1,v_2,v_3. Also find the values of a for which v_1,v_2,v_3 are linearly independent.

    v_1=\begin{bmatrix}1  \\ a \\ a\end{bmatrix} v_2=\begin{bmatrix}a  \\ 1 \\ a\end{bmatrix} v_3=\begin{bmatrix}a  \\ a \\ 1\end{bmatrix}

    When I try to do rref I get all zeros in the bottom...
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  2. #2
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    Quote Originally Posted by Zocken View Post
    Find all values a for which \begin{bmatrix}1  \\ 1 \\ 1\end{bmatrix} is in the span of v_1,v_2,v_3. Also find the values of a for which v_1,v_2,v_3 are linearly independent.

    v_1=\begin{bmatrix}1  \\ a \\ a\end{bmatrix} v_2=\begin{bmatrix}a  \\ 1 \\ a\end{bmatrix} v_3=\begin{bmatrix}a  \\ a \\ 1\end{bmatrix}

    When I try to do rref I get all zeros in the bottom...
    For the first problem you want all a such that x[1 a a]+ y[a 1 a]+ z[a a 1]= [1 1 1] for some numbers x, y, z. That gives the three equations x+ ay+az= 1, ax+y+az= 1, and ax+ay+ z= 1. For what values of a does that have a solution?

    For the second problem you want to look at x[1 a a]+ y[a 1 a]+ z[a a 1]= [0 0 0]. An obvious solution is x=y= z= 0. For what values of a is that the only solution?

    Are you saying you get all zeros in the last row of the augmented matrix, for all a?
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  3. #3
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    yes, when I augment the matrix with 1 1 1 I get the bottom with zeros. Is there something I am missing?
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