Systems of Linear Equations

Summary

A linear system asks for all vectors $\mathbf{x}$ satisfying $A\mathbf{x}=\mathbf{b}$ . The solution set is empty, a single point, or an affine subspace. Matrix form and geometry of intersecting hyperplanes organize the theory.

Prerequisites

High-school systems of equations; Vectors and Dot Product helpful. Hub: Linear Algebra.

Definition / Statement

A linear equation in unknowns $x_1,\dots ,x_n$ has the form

A system of $m$ linear equations is

In matrix form, with $A=(a_{ij})\in {\mathbb{R}}^{m\times n}$ , $\mathbf{x}\in {\mathbb{R}}^n$ , $\mathbf{b}\in {\mathbb{R}}^m$ ,

Objects and Dimensions

Notation

Conditions / Assumptions

• Entries real unless stated otherwise.
• Linear means no products of unknowns, no powers other than $1$ , and no nonlinear functions of unknowns. The equation $A\mathbf{x}+\mathbf{c}=\mathbf{b}$ is still linear in $\mathbf{x}$ after moving $\mathbf{c}$ ; the map $\mathbf{x}\mapsto A\mathbf{x}+\mathbf{c}$ is affine, not linear, if $\mathbf{c}\ne \mathbf{0}$ .

Matrix / Vector Form

Equivalent views of $A\mathbf{x}=\mathbf{b}$ :

1. Row view: each equation is a hyperplane (dot product of a row of $A$ with $\mathbf{x}$ equals $b_i$ ).
2. Column view: $\mathbf{b}$ is a linear combination of the columns of $A$ with coefficients $x_j$ .

Geometric Interpretation

• $n=2$ : lines in the plane; solutions are empty, a point, or a line.
• $n=3$ : planes in space; solutions are empty, a point, a line, or a plane.

If $\mathbf{b}=\mathbf{0}$ (homogeneous system), $\mathbf{x}=\mathbf{0}$ is always a solution; extra solutions form a subspace (the null space of $A$ ).

Procedure

1. Form the augmented matrix $[A\mid \mathbf{b}]$ .
2. Row-reduce (see Matrices and Row Reduction) to row echelon or reduced row echelon form.
3. Read pivot variables and free variables; parametrize the solution set.
4. Declare inconsistency if a row $[0\cdots 0\mid c]$ with $c\ne 0$ appears.

Worked Example

The second equation is twice the first, so the system is consistent with infinitely many solutions: $y=3-x$ , free variable $x=t$ , so $\mathbf{x}=⟨t,3-t⟩$ .

If the second right-hand side were $7$ , the system would be inconsistent.

Common Mistakes

• Calling $y=mx+b$ a linear map in $x$ without rewriting; as a function of one variable with intercept, it is affine.
• Forgetting free variables when rank $A<n$ .
• Concluding uniqueness from $m=n$ without checking invertibility/rank.

Connections

• Related: Matrices and Row Reduction, Determinants, Vector Spaces and Bases
• Next: Matrices and Row Reduction
• Geometry: Lines and Planes

References

Linear systems and solution geometry follow MIT 18.06 and standard introductory linear algebra.^[1]