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Linear algebra-linear combination

  1. Mar 9, 2008 #1
    1. The problem statement, all variables and given/known data
    For each matrix, can you write the third column of the matrix as a linear combination
    of the first two columns?

    [tex]
    \left[ \begin{array}{cccc} 1 & 2 & 3 \\ 7 & 8 & 9 \\ 4 & 5 & 6 \end{array} \right]
    [/tex]

    2. Relevant equations
    x=a(U1)+b(U2)


    3. The attempt at a solution
    I let x equal the third column, U1 as the first column, and U2 as the second column. I solved the augmented matrix and got:

    [tex]
    \left[ \begin{array}{cccc} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} \right]
    [/tex]

    which a=-1, b=2.

    This where I'm confused. Do I just multiply a by the first column and b by the second then that will give me a matrix that is the linear combination wrt the third column?
     
  2. jcsd
  3. Mar 9, 2008 #2
    A linear combination of x and y is a*x + b*y. So you want to know if you can find some a and b such that
    a*first column + b*second column is equal to third column.

    Can you?
     
  4. Mar 10, 2008 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Saying "third column is a linear combination of the first two columns" is the same as saying x+ 2y= 3, 7a+ 8y= 9, 4a+ 5b= 6 for some a, b, c. Can you solve those three equations? One way to solve a system of equations is to set up the "augmented" matrix and row-reduce. Do you see that you have already done that? What are x and y?

    By the way, if the question was really "can you write the third column of the matrix as a linear combination
    of the first two columns?" then you should have been known the answer as soon as you saw the last row.
     
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