Linear Algebra for Machine LearningLinear Algebra for Machine Learning

How your phone gathers photos of the same face, and why the obvious guess is only half right. Zero, one, two, three dimensions, and far beyond. Span and basis, first glimpsed. Why we choose perpendicular rulers. A photo as one point in thirty-six million dimensions, and every possible photo already waiting in that space. Distance in any dimension: Pythagoras, and the taxicab. Learning as a point walking towards the truth. Manifolds, and the manifold hypothesis.

Points become arrows. Notation, from brackets to Dirac’s ket. Scaling, adding and subtracting: the parallelogram, element by element, and king − man + woman. Why vectors of different dimensions cannot be added. The rules underneath, from real analysis, and where floating point quietly breaks one. Linear combinations, the ghost between two photos, and faces that morph. Span, basis and dimension, and first sightings of linear independence and the vector space.

The language of data: samples, features, targets and predictions, and the notation papers assume you already know. The family of products. A x as a mix of the columns of A; a row vector times a matrix as a mix of its rows. Matrix times matrix, one column at a time. The identity, the transpose, and why the shapes must agree. Why vectors are columns. And the engine of every neural-network layer, worked by hand.

Why plain adding fails: units, importance and direction. Two proofs: that a recipe works, and that nothing else does. Linear means a constant rate and no interaction, and why machine learning means linear in the weights. A x = b, and why x and b live in different spaces. The row picture: every recipe at once, ripples of predictions, measurements that tilt and scores that slide. Walking one patient’s line. Hyperplanes in any dimension, and the network that memorised random labels. The column picture: arrows walked tip to tail to reach b. Two spaces, one bridge. Collapsed arrows, scores lifted off the sheet, and why a linear system never has exactly two answers.

From counting rods in the Nine Chapters to Gauss and least squares. Three reversible moves that change the equations but never the answer. The first elimination, seen in both pictures, and the walk from Lecture 4a in disguise. Upper triangles and back substitution. Pivots as units of new information, row exchanges, and the augmented matrix. Every move a matrix, and E A = U. Multiply the pivots: the determinant, as area and volume. Zero pivots that report no answer or infinitely many, and nearly zero pivots that destroy sixteen digits of precision. The recipe, revealed, and a new patient predicted.