I have a function, which maps every point in the 3D space, to an other.
How is it possible, to find a matrix, which works the same, as the function, if I multiply 3D vectors with it?
It's sure, the function can be reproduced using a translate, and a rotation matrix, I just can't figure it, how.
So we have a function $f:\mathbb R^3 \rightarrow \mathbb R^3,$ from which we know that it is an affine transformation, i.e. there are a matrix $A\in \mathbb R^{3\times 3}$ and a vector $b\in \mathbb R^3$ such that $\forall x\in \mathbb R^3:f(x) = Ax+b.$ We want to find $A$ and $b.$
First, we have $f(0) = A0+b = 0+b = b,$ so we have $b.$ Next, let $e_1,e_2,e_3$ be the standard basis of $\mathbb R^3.$ Then, $A$ is the matrix $[Ae_1,Ae_2,Ae3],$ i.e. for $j\in\{1,2,3\},$ the $j$-th column of $A$ is the vector $Ae_j.$ But $$ Ae_j = f(e_j) - b. $$ So we also have $A.$