Systems of Linear Equations and Solutions

Matrix Representation (Ax = b)

A system of linear equations is a collection of two or more linear equations involving the same set of variables. It can be written in a compact form using matrices as:

Ax = b

Where:

• A is an m × n matrix (called the coefficient matrix)
• x is a column vector of variables (n × 1)
• b is a column vector of constants (m × 1)

Example:
Consider the system:
2x + 3y = 5
4x + y = 6

This can be rewritten in matrix form as:

A = [[2, 3], [4, 1]],
x = [x, y]ᵗ,
b = [5, 6]ᵗ

So the matrix equation becomes:
Ax = b

Gaussian Elimination (Row Reduction)

Gaussian elimination is a method used to solve systems of linear equations by transforming the augmented matrix [A|b] into a simpler form called row echelon form (REF) or reduced row echelon form (RREF).

The process consists of the following elementary row operations:

• Row swapping: Interchange two rows
• Row scaling: Multiply a row by a non-zero constant
• Row replacement: Replace a row by the sum or difference of itself and a multiple of another row

The goal is to simplify the system to a form like:
x + ay + bz = c
y + dz = e
z = f
So that variables can be solved by back-substitution.

💡 Gaussian elimination is the backbone of many computer algorithms for solving large linear systems.

Existence and Uniqueness of Solutions

A system of linear equations can have one of the following:

• Unique Solution: If the number of pivot positions equals the number of variables and rank(A) = rank([A|b]), then the system is consistent and has a unique solution.
• Infinite Solutions: If the system is consistent but there are fewer pivot positions than variables, then there are free variables, and the system has infinitely many solutions.
• No Solution (Inconsistent): If rank(A) ≠ rank([A|b]), the system is inconsistent and has no solution.

Rouché–Capelli Theorem: This theorem provides a test for consistency:
A system Ax = b is consistent ⟺ rank(A) = rank([A|b])

💡 Geometrically, this corresponds to whether lines/planes intersect at a point (unique), along a line/plane (infinite), or not at all (no solution).