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# A linear system is equivalent to a vector equation.

Put content here**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.

# Parents

* Linear systems and matrices