Find the matrix of $T$ given :$ T([3, 5]) = [2, -1]$ and $T([1, 2]) = [3, 7]$ , $T$ is linear.

Question

I know that T(v) = v' = Av , where v is the vector, v' is the image, and A is the matrix of transformation

So I've set the two images (v') equal to the matrix [S sub1, S sub 2]

What does [S sub1, S sub2] have to equal to?

I am stuck at this point and I don't know how to further attack this problem.

Answer 1

One approach is to figure out what $T$ does to the two unit vectors $(1,0)$ and $(0,1)$. The first column of $A$ is the vector $T(1,0)$. The second column of $A$ is the vector $T(0,1)$. Notice that $3(1,2)-(3,5)=(0,1)$. So, $$ T(0,1)=3T(1,2)-T(3,5)=3(3,7)-(2,-1)=(7,22). $$ Thus, the second column of $A$ is $\begin{pmatrix} 7\\22\end{pmatrix}$. Do the same for the first column with $(1,0)$.

Answer 2

What you have is $$ A . \left[ \begin{array}{cc} 3 & 1 \\ 5 & 2 \end{array} \right] = \left[ \begin{array}{cc} 2 & 3 \\ -1 & 7 \end{array} \right] $$ Now can you finish?