Definition of matrix equation

Definition: Matrix Equation

A matrix equation is an equation of the form:

Ax = b

where:

• A is an m × n matrix (the coefficient matrix)
• x is an n × 1 column vector (the unknown)
• b is an m × 1 column vector (the constant)

Interpretation: The product Ax is defined as the linear combination of the columns of A using the entries of x as weights:

Ax = x₁·col₁(A) + x₂·col₂(A) + ... + xₙ·colₙ(A)

Example: If A = [[1, 2], [3, 4]] and x = [5, 6]ᵀ, then:

Ax = 5·[1, 3]ᵀ + 6·[2, 4]ᵀ = [5, 15]ᵀ + [12, 24]ᵀ = [17, 39]ᵀ

This definition bridges the algebraic form of linear systems with the geometric concept of linear combinations and span.