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!!

Since the vectors $(4,1)$ and $(1,5)$ are linearly independent (not multiples of each other, in dimension two), you can write the matrix rel the basis that they form simply as $$\begin{pmatrix}a&b\\c&d\end{pmatrix}$$, where $\begin{cases}a(4,1)+c(1,5)=(0,21)\\b(4,1)+d(1,5)=(-19,10)\end{cases}$.
If you wish to get the matrix in the standard basis, conjugate by the change of basis matrix $$\begin{pmatrix}4&1\\1&5\end{pmatrix}$$.
