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Affine Sets Revisited

Now that we have the language of subspaces and dimension, we can revisit the solution structure from Chapter 1. An affine set is a translation of a subspace: $S = \mathbf{p} + V$ for some point $\mathbf{p}$ and subspace $V$ .

The solution set of a consistent system $A\mathbf{x} = \mathbf{b}$ is an affine set. Its dimension equals the nullity of $A$ (number of free variables). When $\mathbf{b} = \mathbf{0}$ , the solution set is a subspace (the null space). When $\mathbf{b} \neq \mathbf{0}$ , it's a shift of the null space.

Affine sets are the linear algebra version of "translated planes" — flat, but not necessarily through the origin.

Formal View

Definition 2.16 — Affine Set

A subset

S \subseteq \mathbb{R}^m

is an affine set if

S = \mathbf{p} + V = \{\mathbf{p} + \mathbf{v} : \mathbf{v} \in V\}

for some subspace

V

and some

\mathbf{p} \in \mathbb{R}^m

. The dimension of

S

\dim(V)

Why This Matters

Affine sets are the geometric shape of all solution sets in linear algebra — understanding them gives immediate structural insight.

Constraint sets in optimization are affine sets
Feasible regions in linear programming are intersections of affine sets (halfspaces)
The set of all images consistent with a set of measurements is an affine set

Learning Resources

Affine sets and convex geometry

MIT OpenCourseWare

Affine sets and their relationship to linear subspaces.

50 min

Quiz

Question 1

The solution set of $A\mathbf{x} = \mathbf{b}$ (with $\mathbf{b} \neq \mathbf{0}$ ) is a subspace.

Common Mistakes

Confusing affine sets (shifted subspaces) with subspaces — subspaces must contain the origin, affine sets need not.