How Do You Determine Linear Transformations in R^2?

[IntroAnalysis]

Homework Statement

If L((1,2)^T) = (-2,3)^T and L((1, -1)T) = (5,2)T determine L((7,5)T)

Homework Equations

If L is a linear transformation mapping a vector V into W, it follows:
L(v1 + v2) = L(v1) +L(v2) (alpha = beta = 1)
and
L (alpha v) = alpha L(v) (v = v1, Beta = 0)

The Attempt at a Solution

I've tried all kind of linear combinations, rotations and reflections, and I just can't get this one. Can someone give a hint?

 

[spamiam]

You first need to write out (7,5) as a linear combination of (1,2) and (1,-1), so you need to find constants a₁ and a₂ such that
$$a_1(1,2) + a_2(1,-1) = (7,5).$$

Can you turn this into a system of simultaneous equations?

 

[IntroAnalysis]

Here is another example. L(x) = vector x +x1 = (x1 + x1, x2 + x1)
So L(1,2)^T = (2, 3)^T and L(2, 3)^T = (4, 5). Therefore, the L(3, 4)^T = (6, 7)^T

We are not taking linear combinations of the first 2 vectors to arrive at the third vector.

 

[spamiam]

I don't really understand your last post. Writing out the vector to which you are applying L as a linear combination of two other vectors works fine in this example with $L(x_1, x_2) = (2x_1, x_1 + x_2)$:

$$a_1 (1,2) + a_2 (2,3) = (3,4)$$

$$a_1 + 2a_2 = 3$$
$$2a_1 + 3a_2 = 4$$

Solving this system, I get a₁ = -1 and a₂ = 2. Then

$$L(3,4) = L(-(1,2) + 2(2,3)) = -L(1,2) + 2L(2,3) = -(2,3) + 2(4,5) = (6,7)$$

as you got before. You can apply the same method to the problem in your original post.

 

[IntroAnalysis]

This is not a linear combination of two vectors. I need to show a linear operator that takes
(1,2) transposed to (-2, 3). And it takes (1, -1) transposed to (5, 2) transposed.

This is not a linear combination of (1, 2) and (1, -1) problem.

 

[Ryker]

It is a linear combination of (1, 2) and (1, -1) problem. If

$$L((1,2)^{T}) = (-2, 3)^{T},$$

then what is

$$L \[ \left( \begin{array}{c} 1\\ 2\end{array} \right)?$$

 

[vela]

This is not a linear combination of two vectors. I need to show a linear operator that takes
(1,2) transposed to (-2, 3). And it takes (1, -1) transposed to (5, 2) transposed.

This is not a linear combination of (1, 2) and (1, -1) problem.

It reduces to the latter problem because the operator L is linear. Take a look again at what spamiam did in his or her last post.

 

[IntroAnalysis]

Thanks for the help.
I see now that 4(1,2)T + 3(1, -1)T = (7, 5)T

Therefore L(7,5)T = 4(-2, 3)T + 3(5, 2)T = (7, 18)T