I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space.
If the transformation is identity, then it is trivial - fixpoints describe the objeced being transformed itself.
In other cases, intuitively, we can either have one fixpoint, a line, or no fixpoints at all. How can I formally show that this is true, other than referring back to vector algebra all the time (a vector in 2D space can either have one crosspoint with another, or they are parallel or they don't have anything in common)
You have an affine transformation $T(v) = Av + w$, and you wish to determine when
$$T(v) = Av + w = v$$
That is,
$$Av = v-w$$
$$Av - v = -w$$
$$(A-I)v = -w$$
The solution space of such a problem is always an affine subspace of your vector space, since it consists of any particular solution plus the kernel of $(A-I)$, which is a vector subspace.