Finding a linear transformation such that $T((4,1)) = (0,21)$ and $T((1,5))=(-19,10)$ Finding a linear transformation such that $T((4,1)) = (0,21)$ and $T((1,5))=(-19,10)$
So, i'm trying to find multiple different ways to do this. I know that every linear transformation can be represented by a matrix, so i'm letting $A$ be a $2 \times 2$ matrix and defining $T(x) = Ax$ and trying to figure out what $A$ has to be. Is this a good way to go about this problem?
I'm ending up with two seperate systems of linear equations, each with four variables and two equations, t hus I have four variables and four equations, so that's good I guess, but I feel that I'm getting lost in the weeds and there may be an easier way to do this...
Would appreciate multiple perspectives on this one... Thank you!!

Let's do this the way you initially tried (and yes, it's a good way to go about the problem). Let
$$A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right].$$
We are given
$$A \left[\begin{array}{c}4\\1 \end{array} \right] = \left[\begin{array}{c} 0\\21 \end{array} \right], \text{ and}$$
$$A \left[\begin{array}{c} 1\\5 \end{array} \right] = \left[\begin{array}{c} -19\\10 \end{array} \right].$$
That is, we have
$$\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[\begin{array}{c}4\\1 \end{array} \right] = \left[\begin{array}{c} 0\\21 \end{array} \right], \text{ and}$$
$$\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[\begin{array}{c} 1\\5 \end{array} \right] = \left[\begin{array}{c} -19\\10 \end{array} \right],$$
giving us the system
$$4a+b=0$$
$$4c+d=21$$
$$a+5b=-19$$
$$c+5d=10.$$
Focussing on the system of two equations with the $a$ and $b$ variables, i.e. the system
$$4a+b=0$$
$$a+5b=-19,$$
we have an easy system of two simultaneous equations with two variables. Solving this, we have $a=1$ and $b=-4$.
Now do the same with the two equations involving the $c$ and $d$ variables, and you're done. Of course, the strategy outlined in the comments is equally valid, but you asked for multiple perspectives!
