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## Homework Statement

Determine if b is a linear combination of a1, a2, and a3.

a1= [1,-2, 0]

a2 = [0, 1, 2]

a3 = [5, -6 8]

b= [2, -1, 6]

## The Attempt at a Solution

Alright, well I used an augmented matrix to solve the problem, and after reducing it completely, the matrix looked like this:

| 1 0 5 |2|

| 0 1 4 |3|

| 0 0 0 |0|

Solving for the c's, I got:

(Where the number to the right of the c denote which one instead of a multiplication number)

c1+ 5c3=2

c2+4c3=3

0c3=0

I found that b is a combination of a1 and a2 if c3 is set to 0, however, since c3 is a free variable, and therefore can be anything, then b wouldn't be a lin. combination if c3 be anything other than 0.

I'm having a hard time understanding this. Is it not a linear combination because there are other possible answers that wouldn't make it so? Or is it a linear combination because atleast one possibility works.

Thank you for any help.