Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?

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

    2. Relevant equations

    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:

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

    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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook