MATH 2310: Linear Algebra with Applications

MATH 2310 is an introduction to linear algebra with a greater emphasis on applications than the other linear algebra course offerings (MATH 2940, MATH 2210). Linear algebra is one of the fundamental mathematical tools in use today, with applications that run far and wide.

Topics covered will include vectors and vector operations, matrices and matrix manipulation, how to solve systems of linear equations, vector spaces and linear transformations, subspaces and orthogonality, determinants, eigenvalues and eigenvectors. Throughout, we will illustrate these concepts and structures with applications.

Textbooks

Introduction to Linear Algebra (5th ed., 2016) by Gilbert Strang

Announcements

• (12/09) There will be some optional review sessions in Malott 203 during the study period for those who want some additional help: Wednesday 2-3pm, Thursday 12-1pm, Friday 12-1pm; please check your email for a more detailed description of the sessions.
• (12/05) The final exam will take place on Saturday, December 14 at 9am in 217 Ives Hall.
• (10/29) The second prelim will take place next Thursday (11/07), in class. Let me know if there any topics of types of questions that you would like me to cover during Tuesday's review session, either in person or via email.
• (09/17) The first prelim will take place next Thursday (09/26), in class. This coming Tuesday will be a day for review, so if there are specific topics or types of questions that you would like me to cover, please let me know either in person or via email.
• The syllabus and the questionnaire from the first day. If you have not done so already, please turn in the latter to me as soon as you can.
• The first class takes place on Thursday, August 29.

Homeworks

• Homework 1 (due Thursday, September 5): Solutions
  • §1.1: 1, 3, 5, 13, 19, 26, 28, 31.
  • §1.2: 4, 6, 8, 12, 16, 27.
• Homework 2 (due Thursday, September 12): Solutions
  • §1.3: 1, 8, 14.
  • §2.1: 5, 8, 9b, 10b, 12, 31.
  • §2.2: 8, 11, 12, 13.
• Homework 3 (due Thursday, September 19): Solutions
  • §2.3: 1, 3, 8, 19, 27, 31.
  • §2.4: 1, 3, 13, 14, 32, 34.
  • §2.5: 2, 5, 7ab, 11, 12, 23, 26, 27, 28, 32, 33.
• There is no homework due on Thursday, September 26, as Prelim I will take place on that day. It will cover material in Strang §1-3, up to and including the material covered by the (09/19) class session. Here are some practice Prelim I's from some previous iterations of the course:
  • Spring 2014
  • Spring 2013
  • Fall 2012
  • Spring 2012
  • Fall 2009
  These may have come slightly earlier or later in the semester than our prelim and with a slightly different version of the textbook, but should give you some idea of what you can be expected to do.
• Homework 4 (due Thursday, October 3): Solutions
  • §2.6: 5, 7, 9, 12.
  • §2.7: 16, 17ab, 20, 39.
  • §3.1: 1, 3, 6, 15, 25, 27, 29.
  • §3.2: 1, 2, 3, 4, 6, 16.
  • §3.3: 1, 5, 23, 34, 36.
• Homework 5 (due Thursday, October 10): Solutions
  • §3.4: 4, 5, 7, 13, 23, 30.
  • §3.5: 1, 3, 4, 8, 11, 12, 15, 21.
  • §4.1: 4, 5, 6, 9, 19, 24, 30.
• Homework 6 (due Thursday, October 17): Solutions
  • §4.2: 1, 2, 5, 6, 7, 11, 12, 13, 17, 23, 24, 27, 30, 31.
  • §4.3: 1, 2, 3, 5, 9 (ignore Figure 4.9b part), 10, 17, 18, 26.
• Homework 7 (due Thursday, October 24): Solutions
  • §4.4: 1, 2, 3, 4, 7, 14, 15, 18, 22, 24.
  • §5.1: 1, 3, 4, 8a, 10, 11, 12, 13 (first matrix only), 14 (first matrix only), 15 (first matrix only), 28.
• Homework 8 (due Thursday, October 31): Solutions
  • §5.2: 1 (first matrix only), 4 (first matrix only).
  • §5.3: 1(a), 6(a), 15, 16.
  • §6.1: 2, 3, 5, 6, 17, 21, 24 (there are only 2), 29, 30.
  • §6.2: 1, 2, 11, 15, 18, 25, 30.
• There is no homework due on Thursday, November 7, as Prelim II takes place on that day. It will cover material up to and including Strang §6.2. Here are some practice Prelim II's from some previous iterations of the course:
  • Spring 2014
  • Spring 2013
  • Fall 2013
  • Fall 2012
  • Fall 2009
  The same caveats for the practice Prelim I's apply here as well (tests are for shorter classes, may be at a slightly different time in the course, with a slightly different version of the book, etc.).
• Homework 9 (due Thursday, November 14): Solutions
  • §5.2: 7, 9, 27, 34.
  • §5.3: 16, 19, 31, 32.
  • §6.1: 19, 33, 34, 35.
  • §6.2: 4, 14, 26, 29, 31, 37, 38.
• Homework 10 (due Thursday, November 21): Solutions
  • §6.3: 4, 5, 8, 11, 13, 15, 21, 26.
  • §6.4: 7, 8, 9, 12, 16, 20, 23, 35.
  • §6.5: 2, 3 (last matrix only), 8, 9, 16, 17, 30, 36.
• Homework 11 (due Thursday, December 5): Solutions
  • §2.4: 35.
  • §10.1: 1, 4, 6, 8, 9, 10, 11, 12, 15, 18.
  • §10.3: 1, 2, 3 (last matrix only), 5, 7, 9, 10.
  • §10.4: 1, 3, 5, 6, 7.
• (Optional) Extra Homework Assignment (due Tuesday, December 10)
• The final exam will take place on Saturday, December 14 at 9:00am, and will have a similar format as the previous exams (e.g. closed book, no calculators, etc.), except that it will be comprehensive (i.e. encompassing content from the whole course) and longer. In particular, the material in the last few classes (e.g. §7 on the SVD), on which we will not have a (required) homework on, is fair game. Here are some final exams from previous iterations of the course.
  • Spring 2014
  • Fall 2013
  • Spring 2013
  • Fall 2012
  Remember the usual caveats: these are from different courses, the content covered may have been slightly different or emphasized different topics, etc.

Topics covered

The contents may be adjusted as the course progresses, in particular, the specific applications, which are in later sections of the textbook.

Basic Objects and Linear Algebraic Operations

• (08/29): Vectors and linear combinations, lengths and dot products (§1.1-1.2)
• (09/03): Matrices, matrix multiplication (§1.3)
• (09/05): Vectors and linear equations, elimination (§2.1-2.2)
• (09/10): Elimination using matrices, matrix operations (§2.3-2.4)
• (09/12): More matrix operations: inverse matrices, elimination = factorization (A = LU) (§2.5-2.6)
• (09/17): Tranposition and its interpretation via linear transformations, (abstract) vectors, subspaces (§2.7, §3.1)
• (09/19): Column spaces, nullspaces, independence, basis, and dimension (§3.2-3.4)
• (09/24): Review
• (09/26): Prelim I, Solutions

Higher Structures in Linear Algebra

• (10/01): Basis and dimension, the four subspaces, towards orthogonality (§3.4-5, §4.1)
• (10/03): Orthogonality, orthogonality of the four subspaces. (§3.5, §4.1)
• (10/08): Projections: onto lines and to general subspaces, projection matrices, relations to geometry and optimization (§4.2)
• (10/10): Least squares approximations, orthonormal bases, and the Gram-Schmidt process (§4.3-4.4)
• (10/15): No class (Fall break)
• (10/17): Orthogonal matrices, QR factorization, introduction to determinants and their properties (§4.4, §5.1)
• (10/22): More on determinants: Cramer's rule, minors (and cofactors), expansion along rows and columns, minors and ranks (§5.2-3)
• (10/24): Eigenvalues and eigenvectors, geometric interpretation (§6.1)
• (10/29): More on eigenvalues and eigenvectors, matrix diagonalization (§6.1)
• (10/31): Powers of matrices, modelling population growth via linear algebra (§6.2)
• (11/05): Review
• (11/07): Prelim II, Solutions

Applications

• (11/12): Solving Systems of Differential Equations with Linear Algebra (§6.3)
• (11/14): Symmetric and Positive Definite Matrices (§6.4-5)
• (11/19): Graphs and Networks (§10.1)
• (11/21): Markov Matrices (§10.3)
• (11/26): Linear Programming (§10.4)
• (11/28): No class (Thanksgiving break)
• (12/03): Introduction to the Singular Value Decomposition (§7.1-2)
• (12/05): "Linear Algebra from 10,000 feet," linear transformations (redux), diagonalization and SVD as finding "good" bases (§8.1-8.2)
• (12/10): Final Review
• (12/14): Final Exam, 9:00am (217 Ives)

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