The subject you now use every day
Linear algebra can feel abstract until you notice how much of daily technology is nothing but matrices multiplying vectors. This lesson connects the machinery you have learned to the places it actually runs — search engines, graphics, data science and machine learning.
Computer graphics
Every rotation, scaling and reflection you see in a game or animation is a matrix multiplying the coordinates of each point — exactly the linear transformations from earlier lessons. A neat trick called homogeneous coordinates adds one extra dimension so that translation (sliding, which is not linear on its own) also becomes a single matrix, letting the graphics pipeline combine rotate-scale-move into one product.
Networks, and Markov chains
A network — web pages, friendships, roads — is stored as a matrix of connections. Model movement through it with a Markov matrix (columns of probabilities), and the long-run behaviour is an eigenvector for eigenvalue 1: the steady state the system settles into.
That is not a toy. Google's original PageRank is precisely this — the steady-state eigenvector of the web's link matrix, ranking a page by how much steady-state "flow" it receives.
PageRank is an eigenvector
The importance of every page on the web was, at Google's founding, the dominant eigenvector of a giant Markov matrix. Everything in this course — matrices, eigenvalues, steady states — meets in that one idea.
Data science, PCA and machine learning
Data is a matrix: rows are examples, columns are features. Principal Component Analysis uses eigenvectors (or the SVD) of that data to find the few directions carrying most of the variation, shrinking huge datasets to a handful of meaningful axes.
Machine learning leans on the same foundation. A neural network is, at heart, a stack of matrix multiplications with simple functions between them; training adjusts the numbers in those matrices. When people say modern AI runs on linear algebra, this is the literal truth.
A worked example
A tiny web of two pages links back and forth with Markov matrix M = [[0, 1], [1, 0]] acting on a probability vector. What steady state does PageRank find?
- The steady state is the eigenvector for λ = 1: solve M v = v.A Markov chain settles into the eigenvector of eigenvalue 1.
- [[0, 1], [1, 0]]·[x, y] = [x, y] gives y = x and x = y.Both equations say the two components are equal.
- So v = [1, 1], normalised to probabilities [0.5, 0.5].Scale so the entries sum to 1.
The steady state is [0.5, 0.5] — the two pages are equally important, found as the λ = 1 eigenvector. That is PageRank in miniature.