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Solution Set Structure

Putting it all together: the solution set of $A\mathbf{x} = \mathbf{b}$ is empty (if inconsistent), a subspace (if $\mathbf{b} = \mathbf{0}$ ), or an affine set (if $\mathbf{b} \neq \mathbf{0}$ and consistent). The dimension of the solution set always equals the nullity of $A$ .

To find the general solution: first find any particular solution $\mathbf{x}_p$ (one specific $\mathbf{x}$ with $A\mathbf{x}_p = \mathbf{b}$ ), then find a basis for $\text{Null}(A)$ . The general solution is $\mathbf{x}_p + c_1\mathbf{n}_1 + \cdots + c_k\mathbf{n}_k$ where $\mathbf{n}_i$ are null space basis vectors and $c_i$ are arbitrary.

This is exactly the same structure as the solution of a linear ODE: general = particular + homogeneous.

Formal View

Theorem 3.16 — Complete Solution Structure

A\mathbf{x} = \mathbf{b}

is consistent with particular solution

\mathbf{x}_p

and

\{\mathbf{n}_1, \ldots, \mathbf{n}_k\}

is a basis for

\text{Null}(A)

, then every solution has the form

\mathbf{x} = \mathbf{x}_p + c_1\mathbf{n}_1 + \cdots + c_k\mathbf{n}_k

for arbitrary

c_1, \ldots, c_k \in \mathbb{R}

. The solution set is

k

-dimensional (

k = \text{Nullity}(A)

Why This Matters

This structure theorem is universal — it describes solution sets in every area of mathematics.

Underdetermined systems (more unknowns than equations) have infinitely many solutions parameterized by free variables
In circuit analysis, loop currents are parameterized by independent loop equations

Learning Resources

Complete solution to $Ax = b$

MIT OpenCourseWare

Complete solution structure: particular + null space.

51 min

Quiz

Question 1

A consistent system $A\mathbf{x} = \mathbf{b}$ with Nullity( $A$ ) = 3 has solution set of dimension:

Common Mistakes

Thinking the dimension of the solution set depends on $\mathbf{b}$ — it only depends on Nullity( $A$ ).