How do I calculate w given the the values of T(w), T(v), T(u), v and u?

Question

I have been given the following information about the linear transformation T:

T(v) = $\begin{matrix} -6 \\ 6 \\ -6 \\ 3 \\ -9 \end{matrix}$

T(u) = $\begin{matrix} -2 \\ 1 \\ 0 \\ -2 \\ 0 \end{matrix}$

where

u = $\begin{matrix} -6 \\ -3 \\ 6 \end{matrix}$

and

v = $\begin{matrix} 6 \\ -3 \\ 10 \end{matrix}$

I also know that

T(w) = $\begin{matrix} 0 \\ 3 \\ -6 \\ 9 \\ -9 \end{matrix}$

The assignment is to calculate the vector w given this information. From this information, I see that the matrix for T is in the form of 5*3 and that if I were to multiply u*'the matrix for T' the result would be T(u), and the same goes for T(v), am I correct this far? Or am I supposed to think differently?

I would be thankful for any help!

Thanks in advance, Nick

Answer 1

Hint

Let $w=\alpha _1u+\alpha _2v$.

Then

$$\alpha _1T(u)+\alpha _2T(v)=T(w),$$ is a system with 5 equations and two unknown variables.