TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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If $A x = b$ has one solution $x_{p}$ , then its full solution set is the affine subspace $x_{p} + N (A)$

Statement

If $x_{p}$ is one solution of $A x = b$ , then $S (A, b) = x_{p} + N (A) .$ Thus every nonempty solution set of a finite linear system is an affine subspace parallel to the nullspace.

Facts & Assumptions

Given: $A \in M_{m \times n} (F)$ , $b \in F^{m}$ and $x_{p} \in F^{n}$ with $A x_{p} = b$ .

[L1]

$S (A, b) = {x : A x = b}$ (Matrix equation $A x = b$ , its solution set, consistency, homogeneous systems and the augmented matrix $[A ∣ b]$ ).

[L2]

$N (A) = {z : A z = 0}$ (Row space, column space, nullspace, row rank, column rank and matrix rank).

[L3]

The multiplication map $L_{A} : x \mapsto A x$ is linear (The rank of a matrix equals the rank of the linear map $x \mapsto A x$ ).

[L4]

The kernel of a linear map is a linear subspace (The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial).

[L5]

An affine subspace is a translate $x + U$ of a linear subspace (Affine subspaces as translates $x + U$ of linear subspaces).

Proof

technique · direct

1.1

If $z \in N (A)$ , then $A (x_{p} + z) = A x_{p} + A z = b + 0 = b$ , so $x_{p} + z \in S (A, b)$ .

L1L2L3algebra

2.1

Conversely, if $x \in S (A, b)$ , then $A (x - x_{p}) = A x - A x_{p} = b - b = 0$ , so $x - x_{p} \in N (A)$ and $x = x_{p} + (x - x_{p}) \in x_{p} + N (A)$ . Thus $S (A, b) = x_{p} + N (A)$ .

step 1.1L1L2L3algebra

3.1

Since $N (A) = \ker L_{A}$ , facts [L3] and [L4] make it a linear subspace. Its translate $x_{p} + N (A)$ is therefore an affine subspace by [L5].

step 2.1L2L3L4L5∎

Depends on

Used by

A consistent underdetermined system has an affine two-parameter solution set Example

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §2.4 (standard reference, not scraped)

If Ax=bAx=b has one solution xpxp​, then its full solution set is the affine subspace xp+N(A)xp​+N(A)