Image of Linear Transformation with Given Vectors

[doublemint]

Homework Statement

Let v1=
1
-2
and v2=
-1
1


Let T:R2R2 be the linear transformation satisfying
T(v1)=
9
7
and T(v2)=
0
-8


Find the image of an arbitrary vector
x
y

Homework Equations

The Attempt at a Solution

I thought it might have to do something with T(u+v)=T(u)+T(v) or some sort of transformation, but I cannot seem to get it...
Any help would be appreciated!
Thanks!

 

[HallsofIvy]

You are given that
$$T\left(\begin{bmatrix}1 \\ -2\end{bmatrix}\right)= \begin{bmatrix} 9 \\ 7\end{bmatrix}$$
and that
$$T\left(\begin{bmatrix}-1 \\ 1\end{bmatrix}\right)= \begin{bmatrix}0 \\ 8\end{bmatrix}$$

And you want to determine
$$T\left(\begin{bmatrix} x \\ y\end{bmatrix}\right$$

Yes, you want to use T(u+v)= T(u)+ T(v). Specifically if $u= Av_1+ Bv_2$ then T(u)= AT(v_1)+ BT(v_2). So first you want find A and B such that
$$\begin{bmatrix}x \\ y \end{bmatrix}= A\begin{bmatrix}1 \\-2\end{bmatrix}+ B\begin{bmatrix}-1 \\ 1 \end{bmatrix}$$

 

[doublemint]

Alright, so I got
A=-x-y
B=-2x-y
I'm guessing then we follow through with T(u)= AT(v_1)+ BT(v_2),

T(x y)=[T(1 -2)T(0 -8)][A B]=[9A, 7A-8B]

Then I sub in A and B:

[9(-x-y), 7(-x-y)-8(-2x-y)]= [-9x-9y, 9x+y]

Is this what I was supposed to do? I think now I have to factor out the x-y, but I can't do it to 9x+y. Did I do something wrong at finding A and B?

 

[doublemint]

I just submitted my work, it was right after all!
Thanks HallsofIvy!