# Homework Help: Linear Operator/transformation Help

1. Oct 22, 2008

### Spraypaint

1. The problem statement, all variables and given/known data
Let L: R^3 -> R^3 be a linear operator. Given that L(1,1,-1)^T = (2,3,4)^T, L(1,-1,1)^T = (1,5,1)^T, and L(-1,1,1)^T = (2,0,0)^T, Compute L(1,2,3)^T. Justify your answer.

2. Relevant equations
None.

3. The attempt at a solution
I honestly do not really have a clue. I have an understanding on linear transformations and how they must satisfy the addition and scalar multiplication, but I don't know how to go about doing this question.

2. Oct 22, 2008

### Dick

If you understand that L is a linear transformation, then you just have to figure out a way to express (1,2,3) in the form a*(1,1,-1)+b*(1,-1,1)+c*(-1,1,1), right?

3. Oct 22, 2008

### e(ho0n3

I assume "^T" mean transpose. You need to write (1,2,3) as a linear combination of the other vectors you were given. Then use the fact that L is linear to compute L(1,2,3).

4. Oct 22, 2008

### Spraypaint

So my understanding of the question is to find how the linear transformation in this problem is defined, and use that to solve for L(1,2,3). Is this wrong?

5. Oct 22, 2008

### Tedjn

You are given how the linear transformation is defined, i.e. the three values. But yes, you will use those definitions along with the fact that L is linear to solve for L(1,2,3).

6. Oct 22, 2008

### Spraypaint

So going on with this idea, you would get three equations ( a + b - c, a - b + c, -a +b +c). Do you set this equal to 1,2,3? Or am I completely off?

7. Oct 22, 2008

### Dick

Yes, you set it equal to (1,2,3) and figure out a, b and c. Then apply L to the sum.

8. Oct 22, 2008

### Spraypaint

Sorry, I think I understand now. I should do a(2,3,4) + b(1,5,1) + c(-1,1,1), and use the a,b,c values I got from the other part to solve for L(1,2,3), right?

9. Oct 22, 2008

### Tedjn

No, you need those at the end in order to get a final result for L(1,2,3).

10. Oct 22, 2008

### Dick

Noooo. You'll need those to get the answer. If v=a*v1+b*v2+c*v3 then L(v)=a*L(v1)+b*L(v2)+c*L(v3). You'll need L(v1), L(v2) and L(v3) to get the final answer.

11. Oct 22, 2008

### Spraypaint

Thank you guys, it makes sense, you surely made my day. =)