Date modified: Wed 2024-07-31 . 12:10 PM
Date created: Wed 2024-07-31 . 12:10 PM
# Math 13 Linear Algebra Homepage SMC Spring 2024 **Welcome to Math 13 Linear Algebra !** (Refresh this page often for updates.) Classes on MW, from 9:30am to 10:50am at MC 3 Office hours: Mondays at 11:00am to 12:00 noon at MC 3, after class **Class held during the Spring 2024 semester, starting from week 1 to week 18.** Some brief notes. Important remark. Just because I type up notes here, this is **not a replacement of lectures**. This is to clarify our material and have a written records of what I have talked about, and additional material that I don't have time to cover in-person. A decline in attendance will result in a decline in notes here. Ok? **Attending lectures, reading the notes, and doing the assigned homework** are the keys to success! Disclaimer. The material is still (largely) based on your textbook Larson (8E) Elementary Linear Algebra, but re-presented and re-organized below. Feel free to reference the textbook if you choose to. [Week 1](#wk1) [Week 2](#wk2) [Week 3](#wk3) [Week 4](#wk4) [Week 5](#wk5) [Week 6](#wk6) [Week 7](#wk7) [Week 8](#wk8) [Week 9](#wk9) [Week 10](#wk10) [Week 11](#wk11) [Week 12](#wk12) [Week 13](#wk13) [Week 14](#wk14) [Week 15](#wk15) [Week 16](#wk16) [Week 17](#wk17) [Week 18](#wk18) [Appendix](#appendix) Week 1. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/00-introductory-remarks|00 Introductory remarks.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/01-system-of-linear-equations|01 System of linear equations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/02-solving-linear-equations-1-echelon-form-and-elementary-row-operations|02 Solving linear equations, a first look: Echelon form and elementary row operations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/03-augmented-matrix-pivot-EF-REF-RREF-row-equivalence|03 Augmented matrix, pivots, EF, REF, RREF, and row equivalence. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/03a-further-understanding-equivalence-relations|03a Further understanding: Equivalence relations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/04-gaussian-elimination|04 Gaussian elimination.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/05-pivots-free-variables-unique-or-parametrized-or-no-solutions|05 Pivots, free variables, unique solutions, parametrized solutions, no solutions. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/05a-more-examples-of-parametrized-solutions|05a More examples of parametrized solutions.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/05b-some-more-applications-and-examples|05b Some more applications and examples.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/06-outcome-of-a-linear-system-and-number-of-solutions|06 Outcome of a linear system, and number of solutions.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/06a-further-understanding-fields-and-other-scalars-of-choice|06a Further understanding: Fields and other scalars of choice.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/06b-further-understanding-solving-system-of-linear-equations-in-different-fields|06b Further understanding: Solving system of linear equations in different fields.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/07-geometric-interpretation-of-simple-linear-systems-and-hyperplanes|07 Geometric interpretation of simple linear systems -- hyperplanes.]] Week 2. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/08-review-of-basic-matrix-terminologies-and-operations-addition-multiplication-and-transpose|08 "Review" of basic matrix terminologies and operations - addition, multiplication, scaling, and transpose. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/09-entrywise-formulas-for-matrix-operations|09 Entry-wise formulas for basic matrix operations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/10-some-special-matrices|10 Some special matrices.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/11-some-basic-matrix-properties|11 Some basic matrix properties.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/12-matrix-vector-product|12 Matrix-vector product.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/12a-application-of-matrix-vector-product-to-matrix-properties|12a Application of matrix-vector product to matrix properties.]] Week 3. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/13-matrix-vector-equation-and-linear-system-of-equations-revisited|13 Matrix-vector equation and linear system of equations revisited.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/14-linear-combinations|14 Linear combinations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/15-span-of-a-set-of-vectors|15 Span of a set of vectors.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/16-solution-sets-of-a-linear-system-Ax-b|16 Structural theorems of solution sets to a linear system $A\vec x = \vec b$.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/16a-proof-of-the-theorem-of-solutions-to-nonhomogeneous-systems|16a Proof of the structural theorem of solutions to nonhomogeneous systems.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/16b-the-fundamental-subspaces-nullspace-and-columnspace|16b The fundamental subspaces: Nullspace and columnspace.]] Week 4. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/17-when-are-two-spans-the-same|17 When are two spans the same?]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/18-spanning-sets-and-removing-redundancies|18 Spanning sets and removing redundancies.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/18a-proof-of-the-algorithm-of-removing-redundant-vectors|18a Proof of the algorithm of removing redundant vectors.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/18b-commentary-on-columnspace-of-a-matrix|18b Commentary on the columnspace of a matrix.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/18c-more-commentary-on-a-spanning-set-ordering|18c More commentary on spanning sets -- On ordering.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/19-linear-independence-a-first-look|19 Linear independence -- a first look.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/19a-proof-of-a-characterization-of-linear-independence|19a Proof of a characterization of linear independence. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/19b-more-examples-of-linear-independence|19b More examples of linear independence. ]] Week 5 - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/20-basis-sets-of-a-set-and-dimension|20 Basis sets of a set, and dimension.]] - Interlude - Data compression and cat pictures. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/21-linear-spaces|21 Linear spaces.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/21a-basis-sets-and-dimensions-of-nullspace-and-columnspace-of-a-matrix.|21a Basis sets and dimensions of nullspace and columnspace of a matrix.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/22-elementary-row-matrices|22 Elementary row matrices.]] Week 6 - **Exam 1 on Monday** - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/23-on-functions|23 On functions.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/23a-bijective-iff-invertible|23a Bijective iff invertible.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/23b-uniqueness-of-inverse-function|23b Uniqueness of inverse function.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/24-linear-transformation-a-first-look|24 linear transformation, a first look]] Week 7 - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/25-left-matrix-multiplication-maps|25 Left matrix multiplication maps.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/26-geometric-transformations|26 Geometric transformations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/27-inverse-of-a-matrix|27 Inverse of a matrix.]] Week 8 - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/28-permutations|28 Permutations.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/29-determinants-part-1|29 Determinants: Part 1 - Defining characteristics.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/30-determinants-part-2|30 Determinants: Part 2 - Computational aspects.]] Week 9 Spring break, no class, but there will be reading. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/31-determinants-part-3|31 Determinants: Part 3 - Applications. ]] ✨✨ Week 10. - Make sure to read notes 31 above. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/32-change-of-basis-part-0-motivation-and-similarity|32 Change of basis: Part 0 - What are linear maps doing and the idea of similarity. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/33-change-of-basis-part-1-coordinate-vectors-ordered-basis|33 Change of basis: Part 1 - Coordinate vectors with respect to an ordered basis.]] Week 11. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/34-change-of-basis-prat-2-matrix-representations-wrt-ordered-basis-sets|34 Change of basis: Part 2 Matrix representations with respect to ordered basis sets.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/35-similarity|35 Similar matrices, determinant and trace.]] Week 12. - **Exam 2 in class Monday !!!!** Covers up to and including notes 35, and any homework assigned this week. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/36-diagonalizability-part-1-eigenvalues-eigenvectors-characteristic-polynomial|36 Diagonalization: Part 1 - Eigenvalues, eigenvectors, characteristic equations.]] Week 13. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/37-diagonalization-part-2-characterizations-eigenbasis-eigenspace-geometric-and-algebraic-multiplicities|37 Diagonalization: Part 2 - Characterization of diagonalizability, eigenbasis, eigenspace, algebraic and geometric multiplicities.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/38-application-to-discrete-dynamical-systems-stochastic-matrices-and-equilibrium-vectors|38 Application to discrete dynamical systems - Stochastic matrices and equilibrium vectors]] Week 14. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/39-application-to-linear-reccurences|39 Application to linear recurrences.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/40-misc-remarks-on-eigenvalues-eigenvectors-and-diagonalization|40 Miscellaneous remarks on eigenvalues, eigenvectors, and diagonalization.]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/41-dot-product-and-orthogonality-in-euclidean-spaces|41 Dot product and orthogonality in Euclidean spaces.]] Week 15. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/42-orthogonal-sets-and-orthogonal-matrices|42 Orthogonal sets and orthogonal matrices. ]] - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/43-orthogonal-projection-and-best-approximation-theorem|43 Orthogonal projection and best approximation theorem.]] Week 16. - No class Monday (Memorial Day) - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/44-least-squares-and-linear-modeling|44 Least squares and linear modeling.]] ✨✨ - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/45-gram-schmidt-process-and-qr-factorization|45 Gram-Schmidt process and QR factorization.]] ✨✨ Week 17. - Last class Monday. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/46-orthogonal-diagonalization-and-real-spectral-theorem|46 Orthogonal diagonalization and real spectral theorem.]] ✨✨ - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/postscript-where-to-go-from-here-after-m13|Post-script -- where to go from here?]] ✨✨ - Review office hours on Wednesday and Friday. Week 18. - **FINAL EXAM MONDAY AT 8AM to 11AM!** Appendix. - [[1 teaching/smc-spring-2024-math-13/linear-algebra-notes/a01-a-brief-word-on-mathematical-statements|A01 A brief word on mathematical statements and proofs.]] - A02 Using computer software to do linear algebra.