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This lecture covers the geometric meaning of linear equations as hyperplanes, introduces Gaussian elimination as the algorithm for solving systems of linear equations (forward elimination → back-substitution, or full reduction to diagonal form), shows how to use GE to invert matrices, and analyses the computational complexity $O(n^3)$. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Understand that each linear equation defines a hyperplane (line in 2D, plane in 3D, etc.)
• [ ] Know the three possible outcomes when solving a system: unique solution, no solution, infinitely many solutions
• [ ] Relate the three outcomes to intersecting hyperplanes (crossing, parallel, overlapping)
• [ ] State the three elementary row operations that preserve solutions
• [ ] Perform Gaussian elimination to transform a system into upper triangular (row echelon) form
• [ ] Solve a triangular system by back-substitution
• [ ] Continue GE backwards to reach reduced row echelon form (diagonal) and read solutions directly
• [ ] Write a system as a matrix equation $Ax = b$ and work with the augmented matrix $(A|b)$
• [ ] Define pivots, row echelon form (REF), and reduced row echelon form (RREF)
• [ ] Identify contradictory rows $(0;0;ldots;0;|;c)$, $c \neq 0$ → no solution
• [ ] Identify zero rows $(0\;0\;\ldots\;0\;|\;0)$ → infinitely many solutions
• [ ] Use GE to invert a matrix by augmenting $A$ with $I$ and reducing to $(I|A^{-1})$
• [ ] Know the computational complexity of GE: $\sim \frac{2}{3}n^3$ operations → $O(n^3)$

1. Hyperplanes and the Geometry of Linear Equations

What is a Hyperplane?

A hyperplane is an $(n-1)$-dimensional subspace embedded in $n$-dimensional space: