Given some implicit equations or a definition of a subset of a vector space, what is the requirement for it to define an affine subspace. I'm looking for something analogous to the test for vector subspaces (closed under adittion and scalar multiplication):
Let $V$ be vector space over a field $\mathcal{K}$ and $S\subseteq V$ a subset, $S$ is a subspace iff $\space \forall s,s'\in S$ and $\forall \lambda \in \mathcal{K}$ we have $s+s' \in S$ and $\lambda s \in S$.
I am aware the definition for an affine subspace is the set $A = a+S$ where $a$ is a "position" vector and $S$ is a vector subspace, I understand this definition. However, I'm under the impression that affine spaces are not in general closed under addition or multiplication.
How do I go about proving for example that: $\{(x_{1}, x_{2}, x_{3}, x_{4}) ∈ \mathcal{K}^4 \space |\space x_{2} = x_{4} = 1, x_{1} + x_{3} = −1\}$ is in fact an affine space, or that $\{v\in \mathbb{R}^n | v_{1}^2+v_{2}^2+... +v_{n}^2=1 \}$ is obviously not since it's the unit sphere?
What is the general procedure?