Given $f: \mathbb R^2\to \mathbb R^2$ where $f((2,7)) = (7,5)$ and $f((1,3)) = (4,1)$ find $f(3,5)$
I assume $f:\Bbb{R^2}\to \Bbb{R^2}$ be a linear map.
Given $f(2, 7) =(7, 5) $ and $f(1, 3) =(4, 1) $
To find $f(3, 5) $ , first we have to write $(3, 5) $ as a linear combination of $(2, 7) $ and $(1, 3)$.
$\begin{bmatrix}2 & 4 \\ 7 & 1 \\\end{bmatrix}\begin{bmatrix}c_1 \\ c_2\\\end{bmatrix}=\begin{bmatrix}3 \\ 5\\\end{bmatrix}$
Solving the above equation, we get $c_1={-4}$, $c_2=11$
Hence, $(3, 5)={-4}(2, 7) +11(1, 3) $
Now, \begin{align}f(3, 5)&={-4}\space f\space (2, 7) +11 \space f\space (1, 3) \\&= {-4}(7, 5) +11(4, 1) \\ &=(16,-9)\end{align}