Linear Systems: Linear combinations

[Novean]

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.

 

[Mark44]

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.

No, this isn't true. c₃ can be set to any value. Of course, this will give different values for c₁ and c₂. The point is that there are lots of linear combinations of a₁, a₂, and a₃ that produce b.

One thing to notice is that if c₃ = 0, then c₁ = 2 and c₂ = 3. What this says is that b = 2a₁ + 3a₂.

Geometrically, b lies in the same plane as a₁ and a₂.

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.

 

[Novean]

Ah! I understand.

Thanks, I forgot to also change the values for c1 and c2 when I picked another free c3.

What had me confused is that the book says it's not a combination. The answer on the back clearly states it in bold. However,the book is probably wrong (Which now a-days I find they often are in regards to answers in the back)

 

[Mark44]

First, make sure you are working the right problem.
Second, if you're working the same problem as in the book, check your answers. In this case, set c₃ to some value (0 is convenient here) and verify that 2a₁ + 3a₂ actually adds up to b.

Try a different linear combination, by setting c₃ to some other value (and calculating values for the two other constants). See if that linear combination of a₁, a₂, and a₃ also ends up with b.