# Linear Combination

• Sep 17th 2006, 11:59 AM
TreeMoney
Linear Combination
I need to find if
3
-1
2

is a linear combination of

1 3 2
-1 1 2
2 -2 -4

I set it up so i had 3 equations they were
3 = x1 + 3x2 +2x3
-1 = -x1 + x2 + 2x3
2 = 2x1 - 2x2 - 4x3

where x1, x2, and x3 are variables. I should be able to solve these and then find the values of the variables and those are the #'s which will make the linear combination possible. But I can't figure it out!!! Help!!!
• Sep 17th 2006, 12:05 PM
ThePerfectHacker
Heir it is solved.
Throught the inverse matrix.
• Sep 17th 2006, 12:37 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Heir it is solved.
Throught the inverse matrix.

So you have changed the task that needs to be done from
solving a set of three linear simultaneous equations to that
of inverting a 3x3 matrix. Other than that you have given the
inverse, so having effectively given the answer, you leave the
OP with a task they are less likely to be able to do.

RonL
• Sep 17th 2006, 12:43 PM
TreeMoney
why did you take the inverse of the matrix? what does that accomplish? I have just started linear algebra and am not sure what you mean or why you would take the inverse. Could you please explain... Thanks
• Sep 17th 2006, 12:48 PM
CaptainBlack
Quote:

Originally Posted by TreeMoney
I set it up so i had 3 equations they were
3 = x1 + 3x2 +2x3
-1 = -x1 + x2 + 2x3
2 = 2x1 - 2x2 - 4x3

Replace the second equation by the second plus the first, give us:

Code:

```3 = x1  + 3.x2 +2.x3 2 =      + 4.x2 +4.x3 2 = 2.x1 - 2.x2 - 4.x3```
Now replace the third row by the third row minus twice the first
row:

Code:

```3  = x1  + 3,x2 +2.x3 2  =      + 4.x2 +4.x3 -4 =      - 8.x2 - 8.x3```
Now we see that the last two rows are multiples of one another,
which leaves us with two effective equations in three unknowns.

This leaves us with multiple solutions so assume a value for x3
and then solve for x1 and x2 in terms of this.

For instance x1=3/2, x2=1/2, x3=0 looks like a solution to me
(assuming no arithmetic errors).

RonL
• Sep 17th 2006, 12:57 PM
CaptainBlack
Quote:

Originally Posted by TreeMoney
why did you take the inverse of the matrix? what does that accomplish? I have just started linear algebra and am not sure what you mean or why you would take the inverse. Could you please explain... Thanks

Ignore it, first he has inverted the wrong matrix, second the matrix
he should have inverted is singular, third even if the matrix was not
singular he is replacing a simple process related to Gaussian elimination
by a more complex problem without showing how to solve the new
problem.

RonL
• Sep 17th 2006, 02:09 PM
TreeMoney
Thanks CaptainBlack and everyone else. I see what I was missing now. Thanks again!!!