Course VI · Open courseware
Linear Algebra
From two lines that cross at the answer to the rotation, stretch and rotation inside every matrix: a first university course in linear algebra, from elimination and the four fundamental subspaces through least squares, determinants and eigenvalues to the singular value decomposition, told as one connected story, in lessons of ten to sixteen minutes.
01
Solving Linear Systems
- 01The Geometry of Linear EquationsOne system, two pictures: intersecting planes, or a combination of column vectors. The column picture is the one that carries the whole course.
- 02Elimination, Matrix Operations, and InversesThe oldest algorithm in the book, made mechanical: pivots knock out unknowns, elimination matrices record every step, and invertibility is exactly 'elimination succeeds'.
- 03LU Factorization, Transposes, and PermutationsElimination, replayed as algebra: A = LU stores the whole computation in two triangles — and transposes and permutations finish the bookkeeping.
02
Vector Spaces and the Four Subspaces
- 04Vector Spaces, Subspaces, and the NullspaceThe subject turns structural: not one solution but the space of all solutions. Ax = 0 defines a subspace — and computing it is elimination again.
- 05The Complete Solution to Ax = bRectangular systems drop the safety net: solutions may not exist, or may come in families. One particular solution plus the whole nullspace is every answer there is.
- 06Independence, Basis, and DimensionHow big is a space, really? Independence trims the redundancy, a basis is just enough vectors, and dimension is the one number that doesn't depend on your choices.
- 07The Four Fundamental SubspacesEvery matrix carries four subspaces — row space, column space, and two nullspaces — with dimensions locked by rank and orthogonality hiding in the pairing. The big picture of the course, on one diagram.
- 08Graphs and NetworksA graph becomes a matrix, and the four subspaces become physics: Kirchhoff's laws are statements about nullspaces.
03
Orthogonality and Least Squares
- 09Orthogonality and ProjectionsPerpendicularity does the heavy lifting: the row space meets the nullspace at a right angle, and projecting onto a subspace is a formula, not a picture.
- 10Least SquaresWhen Ax = b has no solution, solve the nearest problem that does: project b where it can be reached. Fitting a line to data is exactly this.
- 11Gram-Schmidt and A = QROrthonormal bases make everything cheap — projections lose their inverses. Gram-Schmidt manufactures one, and A = QR records the manufacture.
04
Determinants
05
Eigenvalues and Dynamics
- 13Eigenvalues and DiagonalizationSome vectors keep their direction when A acts — and they crack the matrix open: A = SΛS⁻¹ turns matrix powers into scalar powers.
- 14Markov Matrices and Difference EquationsA population shuffles between states, step after step. For a regular Markov matrix the eigenvalue 1 holds the steady state, and the size of the second-largest eigenvalue sets how fast everything else fades.
- 15Differential Equations and the Matrix Exponentialdu/dt = Au is the continuous twin of matrix powers: eigenvalues decide growth, decay, or oscillation, and e^{At} packages every solution at once.
06
Symmetric Matrices, Positive Definiteness, and the SVD
- 16Symmetric and Positive Definite MatricesSymmetry buys the best possible spectral news: real eigenvalues and orthonormal eigenvectors. Positive definiteness adds energy — xᵀAx > 0 — and four equivalent tests for it.
- 17Matrices in EngineeringStiffness matrices from springs and masses: where positive definite matrices come from in the physical world, and why boundary conditions decide singularity.
- 18Similar Matrices and the Singular Value DecompositionSimilarity sorts matrices into families with shared eigenvalues — and the SVD ends the course's factorization story: any matrix at all becomes UΣVᵀ, rotation-stretch-rotation.
07
Linear Transformations and Applications
- 19Complex Matrices, Fourier, and the FFTLet vectors go complex: inner products grow conjugates, the Fourier matrix becomes the most useful matrix in engineering, and a factorization trick makes it fast.
- 20Linear Transformations and Choice of BasisBehind every matrix is a linear map; behind every basis, a different matrix for the same map. Choosing the basis well is what diagonalization and the SVD were secretly doing.
- 21Linear Programming and ComputationThe course's applied tail: optimizing over a polytope of constraints, and what linear algebra looks like when the matrices are huge and the arithmetic is floating-point.