A linear system is equivalent to a vector equation.

Equivalence of Linear Systems and Vector Equations

A system of linear equations can be rewritten as a single vector equation. Consider the system:

a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ

This is equivalent to the vector equation:

x₁[a₁₁, a₂₁, ..., aₘ₁]ᵀ + x₂[a₁₂, a₂₂, ..., aₘ₂]ᵀ + ... + xₙ[a₁ₙ, a₂ₙ, ..., aₘₙ]ᵀ = [b₁, b₂, ..., bₘ]ᵀ

Each column of coefficients becomes a vector, and the solution is a linear combination of these column vectors equal to the constant vector. This perspective reveals the geometric meaning of a linear system: finding scalars that combine the column vectors to produce the target vector.