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Linear algebra-linear combination

  • Thread starter NIZBIT
  • Start date
  • #1
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Homework Statement


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]

Homework Equations


x=a(U1)+b(U2)


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?
 

Answers and Replies

  • #2
16
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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?
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,793
922
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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