How to define affine transformation

Question

Define affine transformation $f: R^2 → R^2$ for which f(0, 0) = (2, 2), f(0, 1) = (3, 4) and f(1, 0) = (4, 3).

How to do this?

Answer 1

the affine transformation you are looking for is of the form $\space x \mapsto Ax+B \space$ with $A \in \mathbb{R^{2 \times 2}} $ and $B \in \mathbb{R^{2}} $ .

With the given coordinates, you can compute the entries of $A$ and $B$ using the equations

$ A \times \begin{pmatrix} 0 \\ 0 \end{pmatrix} + B= \begin{pmatrix} 2 \\ 2 \end{pmatrix} \\ \ A \times \begin{pmatrix} 0 \\ 1 \end{pmatrix} + B = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \\ A \times \begin{pmatrix} 1 \\ 0 \end{pmatrix} + B = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$