Linear Operator/transformation Help

[Spraypaint]

Homework Statement

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.

Homework Equations

None.

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.

 

[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?

 

[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).

 

[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?

 

[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).

 

[Spraypaint]

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?

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?

 

[Dick]

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

 

[Spraypaint]

[edit]

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?

 

[Tedjn]

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

 

[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.

 

[Spraypaint]

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