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.