Linear Algebra — Vectors, Matrices, and the Geometry of Data

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• What a vector is -- arrow, list, and point ~16 min
• Vector addition and scalar multiplication ~16 min
• Vectors in R3 and beyond: Rn ~15 min
• Linear combinations, first look ~15 min
• The dot product ~18 min
• Length, distance, and unit vectors ~16 min
• The angle between two vectors, and Cauchy-Schwarz ~18 min
• Orthogonality ~14 min
• Projection onto a vector, and the orthogonal component ~18 min
• The cross product in R3 ~20 min
• Data as vectors: features, records, and why n is large ~15 min

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• Vector and parametric equations of a line ~18 min
• Lines in R3, and skew lines ~15 min
• Planes: normal vectors and the scalar equation ~20 min
• Parametric equations of a plane ~18 min
• Distance from a point to a line ~18 min
• Distance from a point to a plane ~16 min
• Intersections of lines and planes ~22 min
• Area, volume, and the scalar triple product ~20 min

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• Linear equations and their solution sets ~18 min
• What solving a system means geometrically ~16 min
• The augmented matrix ~15 min
• Row operations and row equivalence ~16 min
• Row echelon form and Gaussian elimination ~20 min
• Reduced row echelon form and Gauss-Jordan ~18 min
• Pivot columns, basic variables, and free variables ~16 min
• Parametric vector form of a solution set ~17 min
• Existence and uniqueness ~15 min
• Homogeneous systems and the structure of solutions ~16 min
• Inconsistent systems, and what the picture looks like ~16 min
• Partial pivoting, and why a computer eliminates differently ~15 min

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• Matrices — notation, shape, and special forms ~15 min
• Matrix addition and scalar multiplication ~16 min
• Matrix-vector multiplication two ways: the row picture and the column picture ~16 min
• Matrix multiplication ~20 min
• Why multiplication is defined that way: composition ~16 min
• Properties, and why AB does not equal BA ~20 min
• The transpose ~16 min
• Identity, diagonal, triangular, and symmetric matrices ~18 min
• Powers of a matrix ~18 min
• Block (partitioned) matrices ~18 min
• The matrix equation Ax = b ~16 min
• Matrices as data, matrices as maps ~15 min

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• The inverse of a matrix ~20 min
• The 2x2 inverse and when it exists ~16 min
• Computing an inverse by row reduction ~20 min
• Properties of inverses ~18 min
• Elementary matrices ~18 min
• The Invertible Matrix Theorem, first installment ~18 min
• Solving Ax = b with an inverse -- and why you usually should not ~18 min
• LU factorization ~20 min
• Solving with LU: forward and back substitution ~20 min
• Permutation matrices, PA = LU, and pivoting ~18 min

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• The 2x2 determinant as signed area ~16 min
• The 3x3 determinant and volume ~22 min
• Cofactor expansion ~22 min
• Determinants of triangular matrices ~15 min
• How row operations change a determinant ~20 min
• Computing a determinant by elimination ~18 min
• Products, transposes, powers, and inverses ~18 min
• Determinant and invertibility ~16 min
• Cramer's rule and the adjugate ~22 min
• Determinant as a scaling factor: area, volume, and orientation ~18 min

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• What a vector space is ~18 min
• Examples beyond Rn: polynomials, matrices, functions ~18 min
• Subspaces ~16 min
• Testing whether a set is a subspace ~18 min
• Span ~18 min
• Span as a subspace, and the column space ~20 min
• The null space ~20 min
• Linear independence ~18 min
• Independence, dependence, and homogeneous systems ~18 min
• Basis ~18 min
• Dimension ~18 min
• Coordinates relative to a basis ~18 min

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• Row space ~18 min
• Column space, revisited ~20 min
• Null space, revisited ~18 min
• The left null space ~18 min
• Bases for all four, from one RREF ~20 min
• Rank ~16 min
• The rank-nullity theorem ~18 min
• The big picture: how the four subspaces fit together ~20 min

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• Functions between vector spaces ~15 min
• What makes a transformation linear ~17 min
• The standard matrix of a linear transformation ~18 min
• Geometry in R2: scaling, shear, rotation, reflection, projection ~20 min
• Composition and matrix multiplication ~18 min
• Kernel ~18 min
• Image (range) ~18 min
• Kernel, image, and rank-nullity again ~16 min
• One-to-one and onto ~20 min
• Invertible transformations ~18 min
• Isomorphisms ~20 min
• Affine maps and homogeneous coordinates ~20 min

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• Coordinate vectors, revisited ~16 min
• The change-of-coordinates matrix ~18 min
• Changing basis in Rn ~22 min
• The matrix of a transformation relative to a basis ~22 min
• Choosing a basis that simplifies the matrix ~18 min
• Similar matrices ~18 min
• Invariants under similarity: determinant, trace, rank ~18 min
• Similarity as a change of viewpoint ~15 min

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• Inner products: generalizing the dot product ~16 min
• Norms -- the L1, L2, L-infinity family ~16 min
• Distance and the triangle inequality ~15 min
• Orthogonal sets and orthogonal complements ~16 min
• Orthonormal sets ~16 min
• Orthogonal matrices ~17 min
• Orthogonal projection onto a line ~18 min
• Orthogonal projection onto a subspace ~18 min
• The orthogonal decomposition theorem ~15 min
• The Gram-Schmidt process ~20 min
• QR factorization ~18 min
• Why orthogonality is the numerically stable choice ~16 min

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• When Ax = b has no solution ~14 min
• The least-squares problem ~18 min
• The normal equations ~18 min
• Least squares by QR ~18 min
• Fitting a line to data ~16 min
• Multiple regression and the design matrix ~18 min
• Residuals, R-squared, and what least squares actually minimizes ~17 min
• Weighted least squares and ill-conditioning ~18 min
• Regularization geometry: what ridge does to the normal equations ~18 min

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• The eigenvector question ~15 min
• Eigenvalues and the characteristic equation ~18 min
• The characteristic polynomial ~18 min
• Eigenspaces ~18 min
• Eigenvalues of triangular and special matrices ~16 min
• Trace, determinant, and the eigenvalues ~14 min
• Algebraic and geometric multiplicity ~16 min
• When the characteristic polynomial has no real roots ~16 min
• Diagonalization ~20 min
• Powers of a matrix by diagonalization ~18 min
• Similarity, diagonalizability, and defective matrices ~20 min
• Symmetric matrices and orthogonal diagonalization ~18 min

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• Markov chains and steady states ~18 min
• PageRank ~20 min
• Discrete dynamical systems and long-run behavior ~18 min
• Population and migration models ~18 min
• Graphs, adjacency matrices, and counting walks ~18 min
• Computer graphics: transforming a scene ~18 min
• Covariance matrices and principal directions ~20 min
• Low-rank structure: when a big matrix is really small ~20 min
• Linear systems of differential equations ~20 min
• What Linear Algebra II adds ~15 min