MATH 223. Linear Algebra.

Section 201, Instructor: Julia Gordon.

Where: CHEM D 200.
When: MWF 10-11am.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: Wednesday 11-12:30, Friday 12:30-2pm, and by appointment.
TA: Yuve; Yuve's office hour: Monday 1-2pm, in MATX 1101 (Math Annex, ground floor).

• Textbook: Janisch, "Linear Algebra". Available online at UBC library
• Course outline
• We will be using Piazza for class discussions. Please ask the mathematical questions on Piazza rather than e-mailing the instructor.
  • The sign-up link .
  • The class link.

Exams

• There will be a "basics quiz" in class in early February (date TBA) and a midterm in mid-March (date TBA).
• The final exam is on April 25, 7-9:30pm (sorry, I have no influence on the time of the exam!). It will be in LIFE 2212.

Announcements

• New due date for the computer project 2 : Friday April 21, 11:59pm on Canvas.


• Office hours after the end of term (in my office, Math 217):
  • Wednesday April 19, 11am-1pm
  • Friday April 21, 1:30pm - 3pm
  • Monday April 24, 11am-1pm.


• The list of topics for the final exam


• The best way to review for the final: try solving the old exams, found here . (Math club sells solution packs, but it is best to solve some of them yourselves. Feel free to post hard problems and questions on Piazza. )
  

HOMEWORK

There will be weekly homework assignments, posted here every Wednesday, and due the following Thursday at 10pm on Canvas.
The solutions are on Canvas, under "Files".

• Here are some resources if you want to start using TeX.

• Problem Set 1 . TeX (source code) for Problem Set 1. (due Monday Jan. 16, on Canvas). A zip file with the questions from Homework 1, helpful if you want to start using Overleaf. In Overleaf, click on "new project", select "import project" and drag this file into the box that opens. This will help you get started right away!
  Supplementary problems on sets. (Do not hand in).


• Problem Set 2 . TeX (source code) for Problem Set 2. (due Wednesday Jan. 25, on Canvas). A zip file with the questions from Homework 2.


• Problem Set 3 . (Due Thursday February 2 on Canvas) TeX source for Problem Set 3 . The zip file with the questions from Problem Set 3 (for easy import into overleaf).


• Problem Set 4 . (Due Thursday February 9 on Canvas) TeX source for Problem Set 4 . The zip file with the questions from Problem Set 4 (for easy import into overleaf).


• Problem Set 5 . (Due Monday Feb 27 at 10pm on Canvas) TeX source for Problem Set 5 . The zip file with the questions from Problem Set 5 (for easy import into overleaf). Note the unusual due date!


• Problem Set 6 . (Due Thursday March 3 at 10pm on Canvas) TeX source for Problem Set 6 . The zip file with the questions from Problem Set 6 (for easy import into overleaf).


• Homework 7: (Due Thursday March 23 on Canvas).
  Two of the problems are from the textbook by Curtis, which can be found At this library link .
  Problem Set 7 . TeX source for Problem Set 7 . The zip file with the questions from Problem Set 7 (for easy import into overleaf).


• Problem set 8: Problems 9.1, 9.2, 9.3, 9.1* and 9.2P from Janisch. (Due Sunday April 2 at 10pm on Canvas -- please note the unusual date!)


• Problem Set 9 . Due Monday April 17 at 10pm on Canvas.
  TeX source for Problem Set 9 . The zip file with the questions from Problem Set 9 (for easy import into overleaf).


• Extra Credit Problem Set . Optional; you can hand in any parts of it by April 23 on Canvas. See the problem set itself for the information on extra credit.
  TeX source . The zip file (for easy import into overleaf).

---

Detailed Course outline

Short descriptions of each lecture and relevant additional references will be posted here as we progress. All section numbers refer to Janisch.

• Monday Jan. 9 : The basics: motivation; starting the language set-up: set notation, subsets, Venn diagrams. Section 1.1 Notes
• Wednesday Jan. 11 : Sections 1.1, 1.2: Cartesian products of sets, maps between sets. Notes.
• Friday Jan. 13 : Diagnostic quiz (not for marks); images/preimages; projections. (Section 1.2). Notes .
  
  • Solution to the diagnostic quiz
  • Just in case, here is a note about negating statements with quantifiers (as you can tell, written based on students' mistakes).
  • Supplementary notes on images and preimages.
  
  Please do Section 1.3 "test" in Janisch.


• Monday Jan. 16 : Vector spaces (Sections 2.1, start 2.3). Notes .


• Wednesday Jan. 18 : Vector spaces, continued; linear subspaces (Sections 2.1, 2.3). Then started complex numbers (Section 2.2) Notes


• Friday Jan. 20 : Complex numbers, and fields in general, including the example of a field of p elements. Sections 2.2 and 2.5. Notes
  A note about reading: Please do Section 2.4 ("test") to make sure you understand all the basic concepts. Section 2.5 is included completely, please read it carefully; Section 2.6 is skipped (read only if you are interested); Sections 2.7 and 2.8 are recommended but completely optional.


• Monday Jan. 23 : Linear span of a set of vectors; linear dependence and independence; the notion of a basis. Sections 3.1 and 3.2. Please read 3.1 (and start 3.2). Notes


• Wednesday Jan. 25 : Bases, continued. The basis extension theorem (Section 3.2). Reading: proof of the basis extension theorem in Section 3.4. Notes .
  The proof in Section 3.4 uses mathematical induction; the proof in class does the same thing without formally referring to induction. Eventually, we will need to use proof by induction. If unfamiliar with induction, please watch short lecture by Prof. Rechnitzer .
  If you like extra practice on induction, here is A worksheet on induction ; solutions to the worksheet problems are in the second half of this lecture note .


• Friday Jan. 27 : Will finish basis exchange lemma (Section 3.2), and prove that the notion of dimension of a vector space is well-defined (section 3.2). The sum and intersection of linear subspaces (section 3.2). Please finish reading the proofs in 3.4. Notes .


• Monday Jan. 30 : Started Chapter 4, Linear transformations. Defined linear transformations and their matrices. Notes.
  Reading: please read Sections 4.1 and 4.2 by Friday!


• Wednesday Feb. 1: Guest lecture by Prof. Patrick Walls . Please bring your computer! Jupyter notebook from today's lecture. (You will need to download it and then import into Jupyter at: ubc.syzygy.ca (after logging in with your CWL).


• Friday Feb.3 : Matrix multiplication. Please read 5.1 and 5.2. Notes .


• Monday Feb.6 : Finishing 4.1-4.2 and 5.1-5.2; Basics quiz. Linear transformations and matrices, continued: the kernel and image of a linear transformation. Rank of a matrix. (We will finish sections 4.1 and 4.2 skipping some of the proofs); Sections 4.4 and 4.5 will be postponed for a while. Notes.
  Solutions to the Basics Quiz .


• Wednesday Feb.8 : Finishing 4.2 and 5.1-5.2: dimesion formula for the kernel and image of a linear transformation. Rank of a matrix. Notes .
  At this point we finished all of Sections 4.1-4.2 and 5.1-5.2. Please read them entirely! Section 4.4 is skipped for now (will do it much later); Section 4.5 -- skipped for now but the computer project is related to it; you can read it for curiosity (not required).


• Friday Feb.10 : Elementary linear transformations (Section 5.3); systems of linar equations (Section 7.1). Notes
  


• Monday Feb.13 : The inverse of a matrix (Section 5.5) Reduced row echelon form of a martix. (Sections 7.1 and 7.3)
  Notes from class .
  This material is not exactly in our textbook (our text does not use the terminology of "echelon form" and "pivots", though the algorithm is described in Section 7.3), but it is very important. Please read, for example, this note: Note by Arash Farahmand . (In fact, there are many many online resources, just google "echelon form of a matrix" and take your pick). The alternate textbook by Curtis also has a section about it, see Chapter 2 section 6 on p.38.
  At this point we finished all of Sections 5.1-5.5 of Janisch; you are responsible for these and (Reduced) Echelon Form, and Sections 7.1, 7.3.
  Section 5.6 (together with 4.5) is optional reading, somewhat covered by our computer project. Will comment on these sections later. You can also read Section 7.5 (optional) -- we will skip it; it could be confusing -- echelon form is enough for solving systems of equations, we do not really need the column operations.


• Wednesday Feb.15 : Did one more example of the reduced row echelon form, and then discussed what it means for a matrix to be invertible, and started Chapter 6, "Determinants". Notes
  Please read 6.1 and 6.2.


• Friday Feb.17 : Determinants. Sections 6.1 and 6.2 Notes .


• FEB 20 - FEB 25 : Break!
  Please remember: the computer project and Homework 5 are due right after the break.


• Monday Feb.27 : Determinants. Sections 6.1 and 6.2, some of 6.3, a little of 6.8 (this will not be tested but it is good to be aware of the Liebniz' formula), 6.9. Notes .
  By now, you should be comfortable with 6.2 and all the statements (if not proofs) in 6.1. The proofs in 6.1 that are skipped by now will be skipped; if interested, please read them and feel free to ask me questions. Next class will cover the rest of 6.3, and then 6.4 and 6.5.


• Wednesday March 1 : Determinants, continued: properties. Determinant of an upper-triangular matrix; determinant of the transpose matrix; det of a block-diagonal (or block upper-triangular) matrix; determinant of the product of matrices. Sections 6.2, 6.3, 6.5. Notes (use "transposition" -- well, not really, just rotation by 90 degrees -- to look at them -- used Landscape mode today).


• Friday March 3 : Cramer's rule; inverting a matrix using the adjugate. Sections 6.4, 6.7 and 7.2. Notes .


• Monday March 6 : Review for the midterm. Please bring questions! Notes .


• Wednesday March 8 : More review. Also covered Section 4.4 (not on the exam!). Notes


• Friday March 10 : Midterm in class.


• Monday March 13 : Euclidean spaces, inner products. Section 8.1 Notes .
  I am sorry, I realized that I was sloppy about the matter of the definition of the angle between vectors (a small correction is posted in the notes, see the blue text; will talk about this next time. Sorry.)


• Wednesday March 15 : Cauchy-Schwarz inequality; norms and triangle inequality; the notion of orthogonal complement (Sections 8.1 and some of 8.2) Notes


• Friday March 17 : Section 8.2: orthogonal projections, Gram-Schmidt orthonormalization process. Notes


• Monday March 20 : Example of using the Gram-Schmidt orthogonalization; orthogonal transformations (Section 8.3). Please read 8.1--8.3 completely.
  Notes
  Section 8.4: optional reading.


• Wednesday March 22 : Eigenvalues and eigenvectors. Sections 9.1 and 9.2. Notes


• Friday March 24 : Sections 9.1 and 9.2, continued. Example of a linear operator that does not have a basis of eigenvalues. Notes .
  Section 9.4 is optional (but recommended) reading.


• Monday March 27 : Finished Sections 9.1 and 9.2. Linear independence of the eigenvectors associated with distinct eigenvalues. Notes.
  Please read again the last page of Section 4.3 (p. 73) about the connection of a matrix of a linear operator with a choice of basis. This point will be discussed again in the lecture. Please also read 11.1.


• Wednesday March 29 : Change of basis formula; transition matrix. Additional reading: a very helpful Wikipedia article .
  Notes


• Friday March 31 : Equivalence of matrices (Section 11.1), Jordan normal form (Section 11.3) and the rank Theorem (Section 11.2) Notes .


• Monday April 3 : Applications of the Jordan normal form -- computing powers of a matrix (see the notes; this is optional material, but you need to know how to compute the n-th powers of a diagonalizable matrix). Started self-adjoint linear operators: see Sections 10.1, 10.2. Notes


• Wednesday April 5 : The Principal Axes Transformation and Spectral Theorem: Section 10.3 (and 11.4). Notes .


• April 12 : An application: discrete dynamical systems, stochastic matrices, steady state: Frobenius-Perron Theorem. This was based on different (excellent) interactive online textbook by Margalit, Rabinoff and Ben Williams. See this section .
  Notes
• THE END. The final exam: April 25. 7pm. :