MATH 223. Linear Algebra.

Section 201, Instructor: Julia Gordon.

Where: CHEM C 126.
When: MWF 10-11am.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: Monday 2:30-3:30pm, Wednesday 12-1pm at MATH 217, and Fridays 6-7pm on zoom (the link posted on Canvas under "annnouncements").
TA: Yuve; Yuve's office hour: Thursdays 4-6pm (between 2 and 4pm also fine) in Math 104.

• 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.

Announcements

• The final exam is on December 17.
• Office hours after the end of term, on zoom (links to be posted on Piazza just before):
  • Tuesday December 12, 3:30-4:30pm.
  • Thursday December 14, 6-7pm,
  • Friday December 15, 8-9pm,
  • Saturday December 16, in person, 11am-2pm, room TBA.


• Deadlines for the last few things: See This post on Piazza.

Review materials for the final exam:

• List of topics for the final
• Final exam from last year
• See also the review problem assignment posted at the end of the "Homework" section. Please look at it even if you do not hand it in.
• 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. )
  
• If you find our textbook hard to read, you might enjoy this excellent interactive online textbook by Margalit, Rabinoff and Ben Williams.
• Older review materials:
  • Basics quiz from last term . Solutions . Please note that ours will be a little harder.
  Here are some problems that were relevant for the midterm from the past exams:
  • Final 2012T1 : Problems 1, 4 from "calculation" section, all problems from "definitions" section, and Problems 1, 2(a) and (b) only, and 3 from Section 3.
  • Exam from 2013 Problems 1, 2 part (2) only, Problem 3 part (1) ("nullity" is the dimension of the kernel), Problem 4, Problem 8, Problem 9.
  • exam from 2008 T2 Problems 2,4,5,6, 8,9
  • Exam from 2008 T1 Problems 1, 6 (a), 7.

HOMEWORK

There will be weekly homework assignments, posted here every Saturday, and due the following Monday at 11:59pm on Canvas.
The solutions will be on Canvas, under "Files".

• Here are some resources for starting to use TeX.

• Problem Set 1 . TeX (source code) for Problem Set 1. (due Monday Sep. 18, 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 Monday September 25, on Canvas). A zip file with the questions from Homework 2.


• Problem Set 3 . (Due Tuesday October 3 on Canvas; note the unusual date because of the holiday on October 2.) 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 Wednesday October 11 on Canvas). Important: ONLY DO PROBLEMS 1--4. IGNORE PROBLEMS 5 and 6 for now. 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 October 23). 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 -- Monday!


• Problem Set 6 . (Due Monday November 6 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 Monday November 20 on Canvas).
  Comments: in the problem 8.2 from Janisch, there is a typo in the matrix A -- it should be symmetric, a_{23} should be 1);
  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 Problem Set 8 . Due Monday Dec 11.
  TeX source for Problem Set 8 . The zip file with the questions from Problem Set 8 (for easy import into overleaf).


• Extra Credit Problem Set -- Harder problems . Optional; you can hand in any part of it by December 16 on Canvas. See the problem set itself for the information on extra credit.
  TeX source . The zip file (for easy import into overleaf).


• The optional easier review problem set (also can be handed in for extra credit by December 15, on Canvas). TeX file ; the zip file .

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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.

• Wednesday Sep. 6 : The basics: motivation; starting the language set-up: set notation, quantifiers. Notes .
  • Aside: check out Russell's Paradox
  • Read: Sections 1.1, 1.2.


• Friday Sep.8 : Subsets, unions, intersections, Venn diagrams. Statements and predicates. Implications. Negation. Approximate notes .
  • Recommended reading about logic: chapters 1 and 2 of "Book of proof" by R. Hammack. pdf file available here .
  • Just in case, here is a note about negating statements with quantifiers (as you can tell, written based on students' mistakes).
  • A supplementary note about unions and intersections of a collection of sets (Quantifiers appear!)
  
  
• Monday Sep. 11 : Logical statements: negating the implication; converse, contrapositive, biconditionals. There are no notes for this part, please read Chepater 2 of R. Hammack's "Book of Proof" (for example).
  Cartesian products of sets, maps between sets (functions). Injective and surjective functions. Sections 1.1, 1.2. Partial notes


• Wednesday Sep. 13 : Images and preimages. Composition of functions. Inverse functions. Approximates notes (though we did not have the "quiz" that appears in these notes. If unsure about surjective functions, try it! solution to that quiz ).
  • Supplementary notes on images and preimages.
    
  • Please do Section 1.3 "test" in Janisch.
  
  
• Friday Sep. 15 : Vector spaces, linear subspaces (Sections 2.1, started 2.3). Notes .


• Monday Sep. 18 : Vector spaces, continued -- easy consequences of the axioms (Section 2.1). Started complex numbers (Section 2.2). Notes


• Wednesday Sep. 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.


• Friday Sep. 22 : 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). Approximate notes


• Monday Sep. 25 : Bases, continued. Preparing for the basis extension theorem (Section 3.2). Notes . Reading ahead: try reading the proof of the basis extension theorem in Section 3.4.
  


• Wednesday Sep. 27 : Proved the basis extension theorem and the basis exchange lemma (Section 3.2), and proved that the notion of dimension of a vector space is well-defined (section 3.2). Please finish reading the proofs in 3.4. Notes .
  The proof in Section 3.4 uses mathematical induction; the proof we will discuss 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 Sep. 29 : The sum and intersection of linear subspaces (section 3.2). The dimension formula. Example of an infinite-dimensional vector space (see p. 47, Section 3.2). Notes .
  At this point we finished Chapter 3. Please do "test" (section 3.3). Sections 3.5 and 3.6 are completely optional; we covered Section 3.4.


• Wednesday Oct. 4 : Chapter 4, Linear transformations. Please read 4.1 and 4.2. Notes


• Friday Oct.6 : Linear transformations and matrices, continued. Associating a matrix with a linear operator. Change of basis matrix. Please read all of 4.1 and 4.2, except we have not yet covered kernels and images of linear transormations, and weill return to them next class. The change of basis matrix is not covered in the textbook, please refer, for example, to a very helpful Wikipedia article . Notes.


• Wednesday Oct. 11 : More about linear transformations and matrices. An important example: a projection operator. Rank-nullity theorem. Notes .


• Thursday Oct. 12 : Finishing 4.2 and 5.1-5.2: dimesion formula for the kernel and image of a linear transformation. Rank of a matrix. Matrix mupltiplication -- sections 5.1, 5.2. 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 Oct.13 : Basics quiz; introduction to Jupyter.


• Monday Oct.16 : Finished matrix multiplication. Started Elementary linear transformations (Section 5.3), and systems of linar equations (Section 7.1). Notes .


• Wednesday Oct.18 : 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.
  


• Friday October 20 : The inverse of a matrix (Section 5.5) Finished the discussion of RREF (reduced row echelon form). Approximate Notes (these notes match the class loosely and have a couple extra exmaples).
  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 and not really recommended) -- 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.


• Monday Oct. 23 : Sections 6.1 and 6.2. Determinants. Note that in class I gave a slightly different definition of the determinant -- instead of demanding that it be "n-linear" (as in the book -- linear in every row vector of the matrix when all the other rows are fixed), I said that we want to specify how it behaves under the elementary row operations. These definitions are equivalent, and the book eventually proves it in one direction -- that their definition implies ours. You can prove the converse as an exercise if you really care (this is completely optional and will not be used in any way).
  Approximate notes .


• Wednesday Oct. 25 : Determinants. Continuing with Sections 6.1 and 6.2, some of 6.3, and 6.8 (this will not be tested but it is good to be aware of the Liebniz' formula), and Section 6.9. Approximate notes .
  The notes contain an extensive "aside" about the Jacobian change of variables formula in multivariable calculus. Please read the notes if interested -- it is not going to be tested in this course though.


• Friday October 27 : Determinants, continued: properties. Determinant of an upper-triangular matrix; det of a block-diagonal (or block upper-triangular) matrix. We also had an extensive discussion of mathematical induction as a method of proof. From now on, you need to be able to make and understand proofs by induction. For review of induction, see the links above (for September 27 lecture). Approximate notes (sorry, the notes are in "landscape mode").


• Monday October 30 -- Wednesday Nov. 1 : Determinant of the product of matrices. Sections 6.2, 6.3, 6.5. Determinant of the transpose matrix. Cramer's rule; inverting a matrix using the adjugate. Sections 6.4, 6.7 and 7.2. Approximate notes for the Monday and Wednesday lectures


• Friday November 3 : Review. Notes (Thanks, Tobias!)


• Monday November 6 : Euclidean spaces, inner products. Section 8.1 Notes .


• Wednesday November 8 : Cauchy-Schwarz inequality; example: the inner product on the space of functions. (Sections 8.1, 8.2). Notes


• Friday November 10 : Midterm in class.


• November 13-15 : Fall break.


• Friday November 17 : 1. norms and triangle inequality; the notion of orthogonal complement (Sections 8.1 and 8.2, continued.) Orthogonal projections, Gram-Schmidt orthonormalization process. Notes


• Monday November 20 : Finishing the the Gram-Schmidt orthogonalization; please also read orthogonal transformations (Section 8.3). Please read 8.1--8.3 completely.
  Notes
  Section 8.4: optional reading.


• Wednesday November 22 : 1. Orthogonal matrices and isometries. (finished 8.3). 2. Eigenvalues and eigenvectors. (Sections 9.1 and 9.2.) Notes


• Friday November 24 : Sections 9.1 and 9.2, continued, and Section 9.4. The characteristic polynomial. The Fundamental theorem of Algebra. At this point we covered all in Sections 9.1-9.4 except for a lemma in 9.1 about linear independence of the eigenvectors corresponding to distinct eigenvalues. We also talked more about the polynomials over the real numbers than the book does (please see today's notes). Notes .
  Section 9.4 is optional (but recommended) reading.


• Monday Nov. 27 : Finished Section 9.1: Linear independence of the eigenvectors associated with distinct eigenvalues. Then will discuss changes of basis. Change of basis formula; transition matrix. Additional reading: a very helpful Wikipedia article .
  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. Please also read 11.1.
  


• Wednesday November 29 : Jordan blocks -- linear operators that do not have a basis of eigenvectors. Jordan normal form (Section 11.3). 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). Please read about Equivalence of matrices (Section 11.1), Notes .


• Friday Dec 1 : Started self-adjoint linear operators: see Sections 10.1, 10.2. The Principal Axes Transformation and Spectral Theorem: Section 10.3 (and 11.4).
  Notes


• Monday December 4 : An example of the recipe for principal axes transformation (see Section 10.3). Then talked about equivalence relations in general, in preparation for talking about equivalence of matrices.
  Notes are coming soon.


• Wednesday December 6: the last class. More on equivalence of matrces: the rank Theorem (Section 11.2), recap of Jordan normal form.
  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: December 17 an noon.