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If $Ax=b$ 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 $Ax=b$ , 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 $Ax_p=b$ .

[L1]

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

[L2]

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

[L3]

The multiplication map $L_A:x\mapsto Ax$ is linear (The rank of a matrix equals the rank of the linear map $x\mapsto Ax$ ).

[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 $A(x_p+z)=Ax_p+Az=b+0=b$ , so $x_p+z\in S(A,b)$ .

L1L2L3algebra

2.1

Conversely, if $x\in S(A,b)$ , then $A(x-x_p)=Ax-Ax_p=b-b=0$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

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