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)
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.