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Linear Systems as Matrix Equations

We can now fully connect the world of linear systems (Chapter 1) to the world of matrices. The system $A\mathbf{x} = \mathbf{b}$ asks: is $\mathbf{b}$ in the column space of $A$ ? If yes, any $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{b}$ is a solution.

Solving a linear system = finding the pre-image of $\mathbf{b}$ under the linear map $A$ . The solution set is $\mathbf{x}_p + \text{Null}(A)$ where $\mathbf{x}_p$ is any particular solution.

This unifies everything from Chapter 1: the trichotomy (no/unique/infinite solutions) corresponds to $\mathbf{b} \notin \text{Col}(A)$ / $\mathbf{b} \in \text{Col}(A)$ and $\text{Nullity}(A) = 0$ / $\mathbf{b} \in \text{Col}(A)$ and $\text{Nullity}(A) > 0$ .

Formal View

Theorem 3.13 — Solvability Criterion

The system

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

is solvable if and only if

\mathbf{b} \in \text{Col}(A)

. When solvable, the complete solution set is

\{\mathbf{x}_p + \mathbf{n} : \mathbf{n} \in \text{Null}(A)\}

for any particular solution

\mathbf{x}_p

Why This Matters

Expressing solvability in terms of column space membership is the geometric key to the theory.

Before solving, check solvability: is $\mathbf{b}$ in the column span?
The particular + homogeneous solution structure is universal — it appears in differential equations too

Learning Resources

Solving $Ax = b$

MIT OpenCourseWare

Complete solution of $Ax=b$: particular solution + null space.

51 min

Quiz

Question 1

If $A\mathbf{x} = \mathbf{b}$ has two solutions $\mathbf{x}_1$ and $\mathbf{x}_2$ , then $\mathbf{x}_1 - \mathbf{x}_2 \in \text{Null}(A)$ .

Common Mistakes

Thinking "any particular solution" means a unique choice — there are infinitely many when Nullity > 0.