Linear Algebra
The mathematics of vectors and the linear transformations that move them — the quiet engine beneath stability analysis, Fourier methods, and network ranking.
Linear Algebra
Linear algebra is the study of vectors and the linear transformations that act on them. Its central insight is that an enormous range of operations — rotating a shape, blurring an image, advancing a physical system by one timestep, ranking web pages — are all the same kind of object: a Matrix that stretches and shears space while keeping straight lines straight and the origin fixed. Once a problem is written this way, a small, reusable toolkit takes over.
That toolkit is why linear algebra sits underneath so much of the rest of this knowledge base. The Stability of an equilibrium is decided by the eigenvalues of a linearized matrix. A Fourier Series is a change of Basis into sinusoids. PageRank is the dominant eigenvector of a link matrix. Learn the vocabulary once and these connections light up everywhere.
The arc of the subject
We build up in three movements:
Objects — a Vector is magnitude and direction; a Matrix packages a transformation as a grid of numbers; the Dot Product measures angle and projection; a Basis is the coordinate frame that turns geometry into numbers.
Actions — a Linear Transformation warps space; Matrix Multiplication composes two warps into one; the Determinant reports how much area or volume the warp scales by.
Structure — the deepest results expose the hidden skeleton of a transformation: Eigenvalues and Eigenvectors are the directions it merely stretches, and the Singular Value Decomposition factors any matrix into a rotation, a scaling, and another rotation.
Why it underpins everything else
Start here
- Vector — magnitude, direction, and components.
- Matrix — a grid of numbers that is secretly a transformation.
- Linear Transformation — how a matrix warps the plane.
- Matrix Multiplication — composing transformations.
- Determinant — the area/volume scale factor.
- Dot Product — projection, angle, and orthogonality.
- Basis — coordinates depend on the axes you pick.
- Eigenvalues and Eigenvectors — the directions a map only stretches.
- Singular Value Decomposition — rotate, scale, rotate.