| Lec # | TOPICS | Readings |
|---|---|---|
| 1 | Systems of Linear Equations | Section 0.1: Systems of linear equations, row equivalence |
| 2 | Echelon Form | Section 0.2: Gaussian and Gauss-Jordan elimination, (reduced) row-echelon form, back-substitution |
| 3 | Matrices | Section 0.3: Matrices, matrix operations, block multiplication |
| 4 | Matrices (cont.) | Section 0.4: Matrices and linear systems, elementary matrices, (reduced) row-echelon matrices |
| 5 | Solution Spaces | Section 0.5: The space of solutions to a homogeneous linear system, uniqueness of the reduced row-echelon form, matrix rank, criterion for existence of solutions |
| 6 | Inverses and Transposes | Section 0.6: Matrix inverses (right, left), invertible matrices, transpose of a matrix, symmetric matrices |
| 7 | Fields and Spans | Section 1.1: Definition of a field F, examples: Q, R, C, Z/pZ (see also 1.6. pp. 132-133), linear combinations of vectors, and spans in Fn |
| 8 | Vector Spaces | Section 1.2: Vector spaces, definition and examples, sub-spaces, the row space, column space, and nullspace of a matrix |
| 9 | Linear Independence | Section 1.3: Linear independent vectors |
| 10 | Basis and Dimension | Section 1.4: Basis of a vector space, dimension, bases for the row space and column space of a matrix, Rank plus nullity theorem for matrices, Basis extension theorem |
| 11 | Coordinates | Section 1.5: Coordinates with respect to an ordered basis, change of coordinates matrix |
| 12 | Review for Quiz 1 | |
| 13 | Quiz 1 (Chapters 0-1) | |
| 14 | Determinants | Section 2.1 (pp. 137-143): Determinant function (definition, properties, uniqueness), computing determinants using row-reduction Section 2.1 (pp. 144-146): invertible matrices, det(AB) = det(A)det(B), det(At) = det (A) |
| 15 | Permutations | Section 2.2: Permutations and the permutation definition of the determinant |
| 16 | Determinants (cont.) | Section 2.2: Permutations and the permutation definition of the determinant |
| 17 | Laplace Expansion | Section 2.3: Cofactor (Laplace) expansion of the determinant, the adjoint of a matrix, finding the inverse using the adjoint, Cramer's rule |
| 18 | Review for Quiz 2 | |
| 19 | Quiz 2 (Chapter 2) | |
| 20 | Linear Transformations | Section 3.1: Linear transformations (definition, examples), matrix associated to a linear transformation |
| 21 | Rank, Kernel, Image | Section 3.2: Properties of linear transformations, rank, kernel, image, Rank plus nullity theorem for linear transformations, one-one, onto, isomorphism |
| 22 | Matrix Representations | Section 3.3: Matrix representations for linear transformations, similar matrices |
| 23 | Eigenspaces | Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |
| 24 | Eigenspaces (cont.) | Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |
| 25 | Diagonalization | Section 3.5: Diagonalizable linear operators and matrices |
| 26 | Cayley-Hamilton Theorem | Section 6.1: Cayley-Hamilton theorem, minimal polynomial |
| 27 | Jordan Canonical Form | Section 6.4 (pp. 373-376): Jordan form, generalized eigenvectors and Primary decomposition theorem from section 6.5 (see also J. Starr's notes from Fall 2004) |
| 28 | Review for Quiz 3 | |
| 29 | Quiz 3 (Chapter 3) | |
| 30 | Computing Generalized Eigenvectors | Section 6.4 (pp. 376-384): More on Jordan form, computing generalized eigenvectors |
| 31 | Norms and Inner Products | Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |
| 32 | Norms and Inner Products (cont.) | Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |
| 33 | Orthogonal Bases | Section 4.2: Orthogonal and orthonormal bases, Gram-Schmidt algorithm, QR decomposition |
| 34 | Orthogonal Projections | Section 4.3: Orthogonal projections, orthogonal complement, direct sums |
| 35 | Isometries, Spectral Theory | Section 4.5 (pp. 282-285): Isometries, orthogonal and unitary matrices |
| 36 | Singular Value Decomposition | Section 4.5 (pp. 286-291): Self-adjoint operators, symmetric and hermitian matrices, eigenvalues of self-adjoint operators, Principal axis theorem, Spectral resolution |
| 37 | Polar Decomposition | Section 4.6: Singular value decomposition, positive (semi)definite matrices, Polar decomposition |
| 38 | Review for the Final |
Table of Contents
Study Materials
Readings
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All of the reading assignments below refer to the main textbook:
Jacob, Bill. Linear Algebra. New York, NY: W.H. Freeman, 1990. ISBN: 0716720310. (Out of print.)
There are two other recommended books for this course:
Hoffman, K., and R. Kunze. Linear Algebra. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1971. ISBN: 0135367972.
Bretscher, O. Linear Algebra with Applications. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2004. ISBN: 0131453343.