Topic 1 of 10

Systems of Linear Equations

The study of linear algebra begins with understanding how to work with systems of linear equations — a set of equations that must all be satisfied simultaneously. Row reduction and matrix representations give us powerful, systematic tools for solving and reasoning about these systems.

1 · Definition

A linear equation in the variables is an equation of the form

where are real (or complex) coefficients and is a constant term. The key property is that each variable appears to exactly the first power and no two variables are multiplied together.

A system of linear equations in unknowns is a finite collection of such equations sharing the same variables:

A solution is a tuple satisfying every equation simultaneously. The solution set is the set of all solutions.

2 · Matrix Form

Every linear system can be written compactly as a single matrix equation , where is the coefficient matrix (), is the column vector of unknowns, and is the column vector of constants:

For the running example:

3 · The Augmented Matrix

To apply row operations we merge and into the augmented matrix , encoding all information of the system in a single array:

For the running example:

4 · Elementary Row Operations

Three reversible transformations leave the solution set unchanged — producing a row-equivalent system:

1. Replacement — Replace row with for any scalar and .
2. Interchange — Swap two rows: .
3. Scaling — Multiply row by any nonzero scalar .

Gaussian elimination applies replacement operations systematically to drive the matrix toward echelon form.

5 · Row Echelon Form

A matrix is in row echelon form (REF) when:

• All zero rows are at the bottom.
• Each row’s leading nonzero entry (pivot) lies strictly to the right of the pivot in the row above.
• All entries below a pivot are zero.

It is in reduced row echelon form (RREF) if additionally every pivot equals and every other entry in a pivot column is . Every matrix has a unique RREF.

6 · Types of Solutions

A linear system falls into exactly one of three cases:

• Unique solution — consistent with no free variables; every variable corresponds to a pivot. In : two lines meet at a single point.

• Infinitely many solutions — consistent with at least one free variable. The solution set is a line, plane, or higher-dimensional affine subspace of .

• No solution — inconsistent. A row of the form with appears — a contradiction.

The number of free variables equals , where counts the pivot positions.

7 · Existence and Uniqueness