MATH 223. Linear Algebra.
Section 201, Instructor: Julia Gordon.
Where: CHEM C 126.
When: MWF 1011am.
My office: Math 217.
email: gor at math dot ubc dot ca
Office Hours: Monday 2:303:30pm, Wednesday 121pm at MATH 217, and Fridays 67pm on
zoom (the link posted on Canvas under "annnouncements").
TA: Yuve; Yuve's office hour: Thursdays 46pm (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 emailing the instructor.
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, 3:304:30pm.
 Thursday December 14, 67pm,
 Friday December 15, 89pm,
 Saturday December 16, in person, 11am2pm, 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.
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 setup: set notation,
quantifiers. Notes .
 Friday Sep.8 :
Subsets, unions, intersections, Venn diagrams.
Statements and predicates. Implications. Negation.
Approximate notes .
 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 ).
 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 welldefined (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 infinitedimensional 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. Ranknullity theorem.
Notes .
 Thursday Oct. 12 :
Finishing 4.2 and 5.15.2: 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 4.14.2 and 5.15.2. 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 5.15.5 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 "nlinear" (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 uppertriangular
matrix; det of a
blockdiagonal (or block uppertriangular) 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 :
CauchySchwarz inequality; example: the inner product on the space of
functions. (Sections 8.1, 8.2).
Notes
 Friday November 10 :
Midterm in class.
 November 1315 : Fall break.
 Friday November 17 :
1. norms and triangle inequality; the notion of
orthogonal complement (Sections 8.1 and 8.2, continued.)
Orthogonal projections, GramSchmidt orthonormalization
process.
Notes
 Monday November 20 :
Finishing the the GramSchmidt orthogonalization; please also
read orthogonal
transformations (Section 8.3).
Please read 8.18.3 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 9.19.4 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 nth powers of a diagonalizable matrix).
Please read about Equivalence of matrices (Section 11.1),
Notes .
 Friday Dec 1 :
Started selfadjoint 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: FrobeniusPerron 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.