Setting Up Linear Systems
Behind every engineering problem with unknowns lurks a system of linear equations. Learning to write it in matrix form — Ax = b — is the first step toward solving anything.
The Ax = b Framework
A system of m equations in n unknowns has a clean matrix representation:
A captures relationships, x holds the unknowns, b is the right-hand side.
Reading the Coefficient Matrix
From equations to matrix: each row = one equation, each column = one unknown.
2x₁ + 3x₂ = 5 and x₁ − x₂ = 1 become exactly this 2×2 system.
Each Equation is a Line
In 2D, every linear equation ax₁ + bx₂ = c describes a straight line. Solving the system means finding where the lines meet.
Planes in 3D Space
In 3D, each equation ax₁ + bx₂ + cx₃ = d defines a plane. Three planes can meet at a point, along a line, or not at all — determining the solution type.
Encoding Everything in [A|b]
Append b as a new column to A:
This single matrix holds all information about the system. Gaussian elimination works directly on [A|b].
Square, Over, and Under
The relative sizes of m and n set expectations — but rank is the true arbiter.
Systems Everywhere
- Electrical circuits: Kirchhoff's laws → linear equations in branch currents
- Structural analysis: force balance at each joint
- Computer graphics: ray-surface intersection
- Data fitting: polynomial through data points (Vandermonde)
- Traffic flow: conservation at each intersection node
Key Takeaways
- Any linear system can be written Ax = b
- A is m×n: m equations, n unknowns
- Geometry: each equation is a hyperplane; solution = their intersection
- Three cases: unique (full rank), none (inconsistent), infinite (rank deficient)
- [A|b] encodes everything for elimination
M3-L2: Gaussian Elimination
We know how to set up Ax = b. Next lesson: how to solve it systematically. Gaussian elimination transforms [A|b] into row echelon form, revealing the solution directly.