How Do You Apply Linear Transformations to Find T(-3, 4)?

[Precursor]

Homework Statement

Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)

Homework Equations

T(x)= b

The Attempt at a Solution

I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?

 

[radou]

The vectors u and v form a basis for R^2, since they're linearly independent (and there is two of them). So, you can write any vector in R^2 as their linear combination, with some coefficients α, β, so, for xome vector x (in your case (-3, 4) ), you have x = α u + β v. Now, what does T(x) equal to?

 

[Precursor]

So I determined α = 3 and β = -2 using your method. I was then able to solve for T(-3,4) = 2.

Is this correct?

 

[radou]

Yes, it is correct.