Definition of matrix representation of a linear system

Definition: Matrix Representation of a Linear System

The matrix representation of a linear system expresses the entire system compactly as:

Ax = b

where:

• A is the m × n coefficient matrix
• x is the n × 1 solution vector (unknowns)
• b is the m × 1 constant vector

This representation is equivalent to both the original system of equations and the corresponding vector equation. The three representations are interchangeable:

1. System of equations: a₁₁x₁ + ... + a₁ₙxₙ = b₁, etc.
2. Vector equation: x₁a₁ + x₂a₂ + ... + xₙaₙ = b (columns of A as vectors)
3. Matrix equation: Ax = b

Why it matters: The matrix representation unifies the treatment of linear systems. Properties of the matrix A (rank, determinant, invertibility) directly determine properties of the solution set (existence, uniqueness, dimensionality).