What a vector actually is

One object, four ways to see it: a list, an arrow, a point, a row of data

Adding, scaling, span

Combine a few vectors, and see exactly which points you can reach

Length, distance, normalization

What "nearest" actually means, and why two honest answers can disagree

Dot product, angle, projection

One number, two ways to compute it, and the operation the rest of deep learning is built on

A matrix is a function

Not a grid of numbers. A rule that moves the whole plane, written in two columns

Matrix–vector multiplication

One row, one neuron, one number: the operation every forward pass runs

Matrix multiplication and the transpose

Two transformations become one matrix, and XYᵀ is every pairwise dot product at once

Rank, collapse and the matrix inverse

Rank counts what survives a transformation, and the inverse exists only when nothing was lost

Tensors, axes and broadcasting

What each axis means, and the rule that decides when mismatched shapes combine — or silently give the wrong answer

Least squares and the normal equation

The fitted values are the projection of y onto the column space of X, and the residual leaves it at a right angle

Change of basis

One vector, two coordinate systems, and the matrix that translates between them

Eigenvectors and eigenvalues

The directions a matrix stretches without turning, and the numbers that say by how much

The covariance matrix and eigendecomposition

The matrix of all pairwise covariances, and why its top eigenvector points along the data

Singular value decomposition and low-rank approximation

Every matrix factors as rotate, stretch, rotate — and keeping the strongest few is the best compression of its size