Intuition behind affine subsets?

Question

I am working through Axler's "Linear Algebra Done Right" and I am having trouble intuiting some of the meaning behind affine subsets. According to 2 exercises in the book we have that

(1) A subset $A$ is affine if and only if for any $v,w\in A$ and any scalar $\lambda$, $\lambda v+(1-\lambda)w\in A$

(2) Given vectors $v_1,...,v_n\in V$ the subset $A$ of $V$ given by $A=\{\sum\lambda_iv_i:\lambda_i\in F,\sum\lambda_i=1\}$ is affine

I more or less understand the definition of affine subsets (they're sort of like subspaces without the identity and they're either disjoint or equal, like equivalence classes) and I more or less understand the mechanics of the proofs of these problems, but I have no intuition for why these conditions imply affine-ness. What's so special about linear combinations the sum of the scalars of which is $1$?

Answer 1

Remember that the straight line through points $w$ and $v$ is given by $$\{w + \lambda (v-w): \lambda\in\mathbb R\}$$ Now a bit of algebra shows that $$w + \lambda(v-w) = w + \lambda v -\lambda w = \lambda v + (1-\lambda) w$$ Or in short, the condition states that a subset is affine if and only if for any two points in that set, the straight line through those two points lies completely in that set.

For example, in three-dimensional space, the affine subsets are the full space (obviously), all planes, all straight lines, the single points (since for $v=w$ we get $\lambda v + (1-\lambda)v = v$) and the empty set.