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Math Help - Linear Combinations

  1. #1
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    Linear Combinations

    A = \[ \left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1 & 4 \\ -1 & 0 & 1 \end{array} \right)\],
    x = \[ \left( \begin{array}{ccc} 1 \\ -2 \\ 3 \end{array} \right)\],
    y = \[ \left( \begin{array}{ccc} 3\\ 0\\ 1 \end{array} \right)\] ,
    z = \[ \left( \begin{array}{ccc} 4\\ -2\\ 5 \end{array} \right)\]

    I need to compute Ax, Ay, Az as linear combinations of the columns of A.

    I'm stuck here, I have tried multiplying x, y and z with A and got:

    x = \[ \left( \begin{array}{ccc} -1 \\ 17 \\ 2 \end{array} \right)\],
    y = \[ \left( \begin{array}{ccc} 7\\ 13\\ -2 \end{array} \right)\] ,
    z = \[ \left( \begin{array}{ccc} 7\\ 34\\ 1 \end{array} \right)\]

    Then it asks me to use the answers to compute the product:

    \[ \left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1  & 4 \\ -1 & 0 & 1 \end{array} \right)\] \[ \left( \begin{array}{ccc} 1 & 3 & 4 \\ -2 & 0  & -2 \\ 3 & 1 & 5 \end{array} \right)\]

    Can someone help me out on what to do?

    I'm really confused...
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  2. #2
    MHF Contributor

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    First you should write those results as "Ax= ", "Ay= ", and "Az= ", not "x= ", etc.

    The columns of A are, of course, \begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}, \begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}, and \begin{bmatrix}1 \\ 4\\ 1\end{bmatrix}

    You want to write [tex]Ax= \begin{bmatrix}-1 \\ 17\\ 2\end{bmatrix}= \alpha\begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}+ \beta\begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}+ \gamma\begin{bmatrix}1 \\ 4\\ 1\end{bmatrix} = \begin{bmatrix}2\alpha+ 3\beta+ \gamma \\ 3\alpha- \beta+ 4\gamma \\ -\alpha+ \gamma\end{bmatrix}
    so you need to solve the three equations 2\alpha+ 3\beta+ \gamma= -1, 3\alpha- \beta+ 4\gamma= 17, and -\alpha+ \gamma= 2 for \alpha, \beta, and \gamma.

    For the second problem, you are simply being asked to multiply two matrices. Don't you know how to do that? You seem to have multiplied a matrix and a vector (column matrix). Just multiply the first matrix times each column in the second matrix, giving the three columns of the product matrix.
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