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The linear algebra that machine learning is written in. The course starts with vectors, dot products and matrix products, then covers elimination and linear systems, determinants, orthogonal projections, eigenvalues, and the singular value decomposition.
01
A vector is a direction with a length. The dot product measures how much two vectors point the same way, and it gives lengths, angles and cosine similarity.
02
A matrix is a linear map, and a matrix product is two maps applied one after the other. Each entry of the product is a dot product of a row and a column.
03
Gaussian elimination turns a system of equations into a triangular one that is easy to solve. LU decomposition stores the steps, and the same steps give the inverse.
04
The determinant is the factor by which a matrix scales area or volume. It is zero exactly when the matrix squashes space flat and has no inverse.
05
Projecting a vector onto a line or a subspace finds its closest point there, and the error is perpendicular. Gram-Schmidt builds an orthonormal basis one projection at a time.
06
An eigenvector keeps its direction when the matrix acts on it, and the eigenvalue is how much it stretches. Power iteration finds the largest one with matrix vector products only.
07
Every matrix is a rotation, a scaling along the axes, and another rotation. Keeping only the largest singular values gives the best low rank approximation.