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MIT 18.06: Linear Algebra

Instructions

MIT 18.06 Linear Algebra is MIT's basic course on matrix theory and linear algebra. It focuses on the topics that other fields use most: systems of equations, vector spaces, determinants, eigenvalues, similarity and positive definite matrices. It is free on MIT OpenCourseWare (Spring 2010 edition), licensed CC BY-NC-SA. Original link https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ Instructor Gilbert Strang, Professor of Mathematics, MIT Main topics (34 lectures plus quiz reviews) - Solving linear systems: the geometry of linear equations, elimination, matrix multiplication and inverses, A = LU - Vector spaces: column space and nullspace, solving Ax = 0 and Ax = b, independence, basis and dimension, the four fundamental subspaces, graphs and networks - Orthogonality: orthogonal subspaces, projections, least squares, Gram-Schmidt - Determinants: properties, cofactor formulas, Cramer's rule, volume - Eigenvalues and eigenvectors: diagonalization, powers of A, differential equations and exp(At), Markov matrices, Fourier series - Special matrices: symmetric and positive definite matrices, complex matrices and the FFT, similar matrices and Jordan form - Singular value decomposition, linear transformations, change of basis and image compression, left and right inverses, the pseudoinverse

Contents · 70

  1. 01-Geometry of Linear EquationsIntroduces solving linear systems in matrix form, using row and column pictures to interpret equations and solutions.
  2. 01-Geometry of Linear EquationsIntroduces solving linear systems through the row and column pictures and matrix form Ax = b, using a two-equation example.
  3. 02-Elimination with MatricesExplains Gaussian elimination, pivots, multipliers, and back substitution for solving linear systems, using matrix operations to transform a system into upper triangular form.
  4. 02-Elimination with Matrices
  5. 03-Multiplication and Inverse MatricesMatrix multiplication entry formulas, dimension rules, and row- and column-based views of how products are formed.
  6. 03-Multiplication and Inverse MatricesExplains matrix multiplication through entry-wise sums, compatible matrix dimensions, and column-by-column and row-based interpretations.
  7. 04-Factorization into A = LUDerives the inverse of a matrix product and of a transpose, then introduces the LU factorization arising from Gaussian elimination.
  8. 04-Factorization into A = LUDerives the inverse of a matrix product and of a transpose, then introduces the LU factorization that connects a matrix to its elimination form.
  9. 05-Transposes, Permutations, Spaces R^nPermutation matrices, row exchanges in Gaussian elimination, the factorization \(PA=LU\), and permutation matrix inverses.
  10. 05-Transposes, Permutations, Spaces R^nCovers permutation matrices for row exchanges and pivoting, the factorization \(PA=LU\), and transpose properties such as \(P^T=P^{-1}\).
  11. 06-Column Space and NullspaceIntroduces vector spaces and subspaces, using planes and lines through the origin, and explains why unions may fail to be subspaces while intersections are subspaces.
  12. 06-Column Space and NullspaceIntroduces vector spaces and subspaces, including closure under linear combinations, examples in R³, and why intersections—but not generally unions—of subspaces are subspaces.
  13. 07-Solving Ax = 0: Pivot Variables, Special SolutionsElimination on rectangular matrices, matrix rank, pivot and free variables, and describing null-space solutions to \(Ax=0\).
  14. 07-Solving Ax = 0: Pivot Variables, Special SolutionsExplains elimination for rectangular matrices, echelon form, pivots and rank, and how free variables describe the null space.
  15. 08-Solving Ax = b: Row Reduced Form RSolving Ax=b by elimination, including augmented matrices, solvability conditions, and the relationship between the column space and row dependencies.
  16. 08-Solving Ax = b: Row Reduced Form RSolvability conditions for Ax=b, augmented-matrix elimination, and the relationship between the column space and row dependencies.
  17. 09-Independence, Basis, and DimensionDefines linear independence and dependence through vector combinations, with examples involving scalar multiples and the zero vector.
  18. 09-Independence, Basis, and DimensionDefines linear independence through zero-sum combinations, with examples of dependent vectors and the role of free variables in homogeneous systems.
  19. 10-Four Fundamental SubspacesIntroduces the four fundamental subspaces—column space, null space, row space, and left null space—and corrects an earlier error about linear independence and rank.
  20. 10-Four Fundamental SubspacesIntroduces the four fundamental subspaces—column space, null space, row space, and left null space—and corrects a basis example by relating row dependence to column dependence.
  21. 11-Matrix Spaces; Rank 1; Small World GraphsMatrix vector spaces and subspaces, with bases and dimensions for all, symmetric, upper triangular, and diagonal 3×3 matrices.
  22. 11-Matrix Spaces; Rank 1; Small World GraphsDimension and bases for matrix spaces and subspaces, including symmetric, upper triangular, and diagonal matrices, and how intersections differ from unions.
  23. 12-Graphs, Networks, Incidence MatricesIncidence matrices encode directed graphs and their electrical-network interpretations; loops create linear dependencies among rows.
  24. 12-Graphs, Networks, Incidence MatricesIntroduces graph incidence matrices, showing how edge directions encode network structure and how loops create linear dependencies. Connects graphs to electrical and other applied networks.
  25. 13-Quiz 1 ReviewReviews the Ax=b portion of a linear algebra course, emphasizing rectangular matrices, vector spaces, the four fundamental subspaces, rank, null spaces, and row reduction.
  26. 13-Quiz 1 ReviewReviews Ax = b topics, emphasizing rectangular matrices, rank, null spaces, the four fundamental subspaces, and row reduction.
  27. 14-Orthogonal Vectors and SubspacesDefines orthogonality for vectors using the dot product, and connects it to vector lengths and the Pythagorean theorem. It introduces the idea of orthogonal subspaces.
  28. 14-Orthogonal Vectors and SubspacesDefines orthogonal vectors using the dot product, relates orthogonality to the Pythagorean theorem, and begins applying the concept to subspaces.
  29. 15-Projections onto SubspacesDerives the projection of a vector onto a line using orthogonality, then expresses the projection as a matrix and examines its behavior.
  30. 15-Projections onto SubspacesDerives the formula for projecting a vector onto a line and the corresponding projection matrix.
  31. 16-Projection Matrices and Least SquaresDerives the projection matrix and its geometric meaning, then begins applying it to find a best-fit straight line.
  32. 16-Projection Matrices and Least SquaresProjection matrices onto a column space and its orthogonal complement; least-squares best-fit lines for data points.
  33. 17-Orthogonal Matrices and Gram-SchmidtOrthonormal bases, orthogonal matrices, and their key properties, including QᵀQ = I and Qᵀ = Q⁻¹ for square matrices.
  34. 17-Orthogonal Matrices and Gram-SchmidtOrthonormal bases, the identity \(Q^mathsf{T}Q=I\), and square orthogonal matrices whose transpose is their inverse.
  35. 18-Properties of DeterminantsIntroduces the determinant through its defining properties: normalization, sign reversal under row swaps, and linearity in a row; connects determinants to invertibility and the 2×2 formula.
  36. 18-Properties of DeterminantsIntroduces determinant properties: normalization, sign changes under row swaps, and linearity in a row, connecting determinants to invertibility and eigenvalues.
  37. 19-Determinant Formulas and CofactorsDerives the determinant formula by multilinearity, identifying the surviving terms as products indexed by permutations, and introduces the cofactor formula.
  38. 19-Determinant Formulas and CofactorsDeriving determinant formulas from linearity and row operations, introducing cofactors, and applying the formulas to matrices with many zeros.
  39. 20-Cramer's Rule, Inverse Matrix, and VolumeDerives the inverse-matrix formula using cofactors and the determinant, and verifies it through matrix multiplication.
  40. 20-Cramer's Rule, Inverse Matrix, and VolumeDerives the inverse-matrix formula using cofactors, then applies determinants to Cramer’s rule and volume.
  41. 21-Eigenvalues and EigenvectorsDefines eigenvalues and eigenvectors through the equation AX = λX, relates zero eigenvalues to the null space, and examines eigenvectors of projection matrices.
  42. 21-Eigenvalues and EigenvectorsDefines eigenvalues and eigenvectors through the equation Ax = λx, relates zero eigenvalues to the null space, and illustrates eigenvectors of projection matrices.
  43. 22-Diagonalization and Powers of ADiagonalizing a matrix using its eigenvectors and eigenvalues, and using the factorization to study powers of the matrix.
  44. 22-Diagonalization and Powers of ADerives the diagonalization formula A = SΛS⁻¹, where S contains independent eigenvectors and Λ contains their eigenvalues.
  45. 23-Differential Equations and exp(At)Solving constant-coefficient linear systems using eigenvalues and eigenvectors, matrix exponentials, and steady-state behavior.
  46. 23-Differential Equations and exp(At)Solving constant-coefficient linear differential equation systems using matrix eigenvalues and eigenvectors, with emphasis on exponential solutions, steady states, and a worked two-variable example.
  47. 24-Markov Matrices; Fourier SeriesMarkov matrices, their eigenvalues and eigenvectors, and how repeated matrix powers approach a steady state; introduces Fourier series as an application of projections.
  48. 24-Markov Matrices; Fourier SeriesExplains how Markov matrices govern repeated state transitions, why 1 is an eigenvalue, and how eigenvalues and eigenvectors determine steady states.
  49. 25-Quiz 2 ReviewReviews orthogonality, least squares, determinants, and eigenvalues, then works through projection matrices and their eigenvalues.
  50. 25-Quiz 2 ReviewReviews orthogonality, least squares, Gram–Schmidt, determinants, and eigenvalues, then works through projection matrices and their eigenvalues.
  51. 26-Symmetric Matrices and Positive DefinitenessExplains the spectral theorem for real symmetric matrices: their eigenvalues are real, and their eigenvectors can be chosen orthonormal, giving the factorization A = QΛQᵀ.
  52. 26-Symmetric Matrices and Positive DefinitenessExplains the spectral theorem for real symmetric matrices: their eigenvalues are real and their eigenvectors can be chosen orthonormal, giving the factorization A = QΛQᵀ. Also introduces positive definiteness.
  53. 27-Complex Matrices; Fast Fourier TransformComplex vectors and matrices, Hermitian transpose, and inner products and lengths in complex space; introduces the Fourier matrix and the Fast Fourier Transform.
  54. 27-Complex Matrices; Fast Fourier TransformIntroduces complex vectors and matrices, Hermitian transpose, and inner products, then explores the Fourier matrix and the fast Fourier transform.
  55. 28-Positive Definite Matrices and MinimaTests for positive definiteness using eigenvalues, determinants, pivots, and the quadratic form xᵀAx, with examples and links to semidefiniteness.
  56. 28-Positive Definite Matrices and MinimaTests and meaning of positive definiteness, including eigenvalues, determinants, pivots, quadratic forms, and the geometry of ellipses and hyperbolas.
  57. 29-Similar Matrices and Jordan FormPositive definite matrices: inverses and sums remain positive definite, while \(A^T A\) is positive semidefinite and is positive definite when \(A\) has full column rank.
  58. 29-Similar Matrices and Jordan FormExplains positive definite matrices, including inverses, sums, and the role of \(A^T A\) in least squares; introduces similar matrices.
  59. 30-Singular Value DecompositionIntroduces the singular value decomposition, its orthogonal bases and stretching values, and its connection to symmetric positive-definite matrices.
  60. 30-Singular Value DecompositionIntroduces the singular value decomposition, its orthogonal bases and singular values, and how it relates to symmetric positive definite matrices.
  61. 31-Linear Transformations and Their MatricesDefines linear transformations through preservation of vector addition and scalar multiplication, with projections as examples and shifts and vector length as non-examples.
  62. 31-Linear Transformations and Their MatricesIntroduces linear transformations as coordinate-free maps, explains projections and the linearity rules, and contrasts them with shifts and vector-length maps.
  63. 32-Change of Basis; Image CompressionExplains how change of basis enables lossy image compression, using pixel data and JPEG as examples.
  64. 32-Change of Basis; Image CompressionHow change of basis enables lossy image compression, including JPEG’s use of correlated pixel values and transform coefficients.
  65. 33-Quiz 3 ReviewReviews Chapter 6 topics: eigenvalues and eigenvectors, symmetric and positive definite matrices, similar matrices, singular value decomposition, and solving linear differential equations using eigenvalues.
  66. 33-Quiz 3 ReviewReviews Chapter 6 topics: eigenvalues and eigenvectors, differential equations, symmetric and positive definite matrices, similar matrices, and the singular value decomposition.
  67. 34-Left and Right Inverses; PseudoinverseReviews the four fundamental subspaces and explains how rank determines when a matrix has a two-sided inverse or a left inverse, including the role of \(A^\mathsf{T}A\) in least squares.
  68. 34-Left and Right Inverses; PseudoinverseExplains matrix inverses for full-rank and full-column-rank matrices, including rank, null spaces, and the invertibility of AᵀA.
  69. 35-Final Course ReviewReviews linear algebra exam problems on rank, column and null spaces, uniqueness of solutions, and invertibility and positive definiteness of matrix products.
  70. 35-Final Course ReviewReviews linear algebra exam topics, including rank, solvability, null and column spaces, and properties of \(A^T A\) and \(AA^T\).