This is the webpage for MAT247 during Winter 2019. All the course documents will be posted here. We will be using Quercus for the purposes of announcements and recording grades.

This is a second course in linear algebra aimed at students in the Math Specialist program.

** Textbook: ** Linear Algebra by Friedberg, Insel, and Spence, fourth edition

The following topics will be covered:

- Diagonalization (Chapter 5 of the textbook)
- Jordan and rational canonical forms (Chapter 7 of the textbook)
- Inner product spaces (6.1-6.6 of the textbook)
- Bilinear forms (6.8 of the textbook)
- Multilinear algebra (time-permitting, reference to be given later)

** Instructor: ** Payman Eskandari

Office location: HU1012 (located on the 10th floor of 215 Huron Street). Please note that the elevators in the building only go up to the 9th floor. From there you have to take the stairs.

Email address: payman@math.utoronto.ca

** Office hours: ** Mondays 1-3 in HU1012

** TAs: **

- Ismail Abouamal (ismail@math.toronto.edu)
- Ozgur Esentepe (ozgur.esentepe@mail.utoronto.ca)

** TA office hours:**

- Ismail: Wednesdays 1:30-2:30 in HU1027
- Ozgur: Thursdays 3:30-4:30 in HU1027

Note: The assignment pdfs below also include some practice problems.

Assignment 1 due Friday Jan 18 (to be submitted on Crowdmark) Solutions

Assignment 2 due Friday Jan 25 (to be submitted on Crowdmark) Solutions

Assignment 3 due Saturday Feb 2 (to be submitted on Crowdmark) Solutions

Assignment 4 due Saturday Feb 9 (to be submitted on Crowdmark) Solutions

Assignment 5 due Friday Feb 22 (to be submitted on Crowdmark) Solutions

Assignment 6 due Saturday March 9 (to be submitted on Crowdmark) Solutions

Assignment 7 due Saturday March 16 (to be submitted on Crowdmark) Solutions

Assignment 8 due Saturday March 23 (to be submitted on Crowdmark) Solutions

practice problems on the material of Week 10 (not to be handed in): exercises 1,2,4,6, 7, 8, 10, 11, 12, 13a (we proved 13b-d in class), 14, 15, 17, 19, 20, 21, 22, 23 of 6.2 of the textbook. (Exercise 23 gives an example that shows that in the result V=W⊕W^{⊥}, the hypothesis of finite-dimensionality of W is essential.)

Assignment 9 due Friday April 5 (to be submitted on Crowdmark) Solutions

midterm solutions (regular sitting)

Reading: 5.1 of the textbook

What we did:

- Tuesday: We reviewed some material from Chapter 2, including coordinate maps, matrix representation of a linear map, change of coordinate matrix, and how the matrix representations of a map for different choices of bases are related to one another.
- Thursday: We covered 5.1.

Next week's plan: Cover 5.2 and some of Appendix E. If there is time, start 5.4.

Reading: Section 5.2 and up to Corollary 2 of Appendix E

What we did:

- Tuesday: We spent most of the lecture discussing problems 4 and 5 from the assignment. In addition, (1) we discussed that the constant term of the characteristic polynomial of a map is just the determinant of the map, and (2) we considered the matrix of rotation by theta, and saw that the matrix is diagonalizable over the complex numbers, but if theta is not an integer multiple of pi, it is not diagonalizable over the reals.
- Thursday: In short, we covered some (most) of 5.2 and some of Appendix E. To be more explicit, we gave three equivalent definitions for a sum of subspaces to be direct, and proved that the sum of eigenspaces corresponding to distinct eigenvalues is direct. As a corollary we deduced that if the sum of the dimensions of the eigenspaces equals the dimension of V (the domain of our map), then the map is diagonalizable. We stated that the converse is also true (proof to be given next Tuesday). We then talked a bit about polynomials over a field, and proved the division algorithm for polynomials ("long division", Theorem E1 of Appendix E).

Next week's plan: Wrap up 5.2 on Tuesday, work on 5.4 on Thursday.

Reading: Section 5.2, the two corollaries after Theorem E1 of Appendix E, some of Section 5.4 (the part on T-cyclic subspaces and the proof of the Cayley-Hamilton still left)

What we did:

- Tuesday: We continued our discussion about polynomials over a field: we proved the two corollaries after Theorem E1 of Appendix E, defined the notion of multiplicity of a root and what it means for a polynomial to split over a field. We observed that if a linear operator has a triangular form, its characteristic polynomial must split over the field. We also showed that in general, the dimension of an eigenspace is bounded from above by the multiplicity of the corresponding eigenvalue (which by definition is the multiplicity of the eigenvalue as a root of the characteristic polynomial).
- Thursday: We finished 5.2. The main result was the equivalence of the following statements for a linear operator T on a finite-dimensional vector space V over a field F:
- (i) T is diagonalizable.
- (ii) The sum of the dimensions of the eigenspaces of T is equal to the dimension of V.
- (iii) V decomposes as the direct sum of the eigenspaces of T.
- (iv) The characteristic polynomial of T splits over F and the dimension of every eigenspace is equal to the multiplicity of the corresponding eigenvalue.

Next week's plan: Discuss cyclic subspaces and prove Cayley-Hamilton on Tuesday. Probably start Chapter 7 on Thursday. (We'll come back to Chapter 6 after we do 7.)

Reading: the rest of 5.4, a bit of 7.1 (the definition of Jordan canonical form and the statement of Corollary 1, which is the main theorem of this part of the course)

What we did:

- Tuesday: We discussed T-cyclic subspaces and proved Cayley-Hamilton.
- Thursday: We started Chapter 7 (we'll come back to Chapter 6 later). We defined Jordan blocks and matrices. We defined the notions of a Jordan basis and Jordan canonical form of a linear operator. We stated the main theorem of 7.1 and 7.2, i.e. that if the characteristic polynomial splits over the field, then the operator has a Jordan canonical form, which is unique up to rearrangement of the Jordan blocks. We then gave a rough outline of the proof of existence of a Jordan canonical form. (In the first part, we will decompose our vector space as a direct sum of invariant subspaces K
_{λ}(which are to be defined), indexed by the eigenvalues, such that the characteristic polynomial of the restricted map to K_{λ}is (t-λ)^{m}, where m is the multiplicity of λ. In the second part of the proof then we will show that each K_{λ}has a Jordan basis.) We then switched gears for the remainder of the lecture and talked about nilpotent maps. We proved that if T is a nilpotent map on an n-dimensional vector space, then the characteristic polynomial of T is (-t)^{n}. Combining with Cayley-Hamilton, we deduced that the nilpotency index of T (i.e. the smallest nonnegative integer k such that T^{k}=0) is at most n.

Next week's plan: Prove existence of Jordan canonical form. If there is time, start 7.2.

Reading: 7.1 (up to Theorem 7.6), rest of Appendix E (Theorems E2, E6-E9)

What we did:

- Tuesday and the first half of Thursday: We defined the notion of a generalized eigenspace of a linear operator T on a vector space V. (For any scalar λ, the generalized eigenspace corresponding to λ is defined as K
_{λ}:={v∈ V: there exists an integer k such that (T-λI)^{k}(v)=0}. The nonzero elements of K_{λ}are called generalized eigenvectors corresponding to λ.) We observed that K_{λ}is not zero if and only if E_{λ}is not zero (i.e if and only if λ is an eigenvalue of T), and moreover, if V is finite-dimensional, then K_{λ}=ker(T-λ I)^{N}for sufficiently large N. We also saw that K_{λ}is invariant under T (and hence, under f(T) for every polynomial f(t)). We then proved the following result, which summarizes the first part of the proof of existence of a Jordan canonical form.Theorem (*): Let T be a linear operator on a finite-dimensional vector space V over a field F.

- (a) If λ is an eigenvalue of T of multiplicity m, then dim(K
_{λ})=m, the characteristic polynomial of the restriction of T to K_{λ}is (-1)^{m}(t-λ)^{m}, and K_{λ}=ker(T-λI)^{m}. - (b) The sum of generalized eigenspaces corresponding to distinct eigenvalues is direct.
- (c) If the characteristic polynomial of T splits over F, then V=⊕K
_{λ}, where the sum is over the eigenvalues of T.

_{λ}). Then the restriction of T-λI to K_{λ}is nilpotent, so that its characteristic polynomial is (-t)^{d}. So the characteristic polynomial of the restriction of T to K_{λ}is (-1)^{d}(t-λ)^{d}. Let S be the map induced by T on the quotient V/K_{λ}. Let f(t) be the characteristic polynomial of S. Then the characteristic polynomial of T factors as (-1)^{d}(t-λ)^{d}f(t). We showed that λ is not an eigenvalue of S, so that f(λ) is not zero. Hence m=d. This proves the first two assertions. The second assertion together with Cayley-Hamilton implies the third.To prove (b), we first showed that if μ≠λ, then the restriction of T-μI to K

_{λ}is injective. Then we used this to show that if ∑v_{λ}=0 (the sum over λ and v_{λ}in K_{λ}), then v_{λ}=0 for all λ. Statement (c) follows easily from (a) and (b). - (a) If λ is an eigenvalue of T of multiplicity m, then dim(K
- second half of Thursday: We started the second part of the proof of existence of a Jordan canonical form. We closely follow the textbook here. We introduced the notion of a cycle of generalized eigenvectors corresponding to an eigenvalue. We observed that a basis of V is a Jordan basis if and only if it is a disjoint union of cycles of generalized eigenvectors. Our goal now is to show that for each eigenvalue λ, the space K
_{λ}has a basis which is a disjoint union of cycles; in view of (c) of the Theorem (*) above, this will finish the proof of existence of a Jordan canonical form. We stated a lemma about cycles of generalized eigenvectors (Theorem 7.6 of the textbook): if the γ_{i}are cycles of generalized eigenvectors corresponding to λ such that their initial vectors are distinct and linearly independent, then the γ_{i}are pairwise disjoint and their union is linearly independent. We did not have time to prove the lemma. (Note that following the textbook, by the initial vector in a cycle we mean the vector that is written first in the cycle, i.e. the only eigenvector in the cycle. The generator of the cycle is referred to as the end vector.)

Next week's plan: Finish the proof of existence of a Jordan canonical form on Tuesday. Talk about uniqueness and how to calculate the Jordan canonical form and a Jordan basis (7.2 of the textbook) on Thursday.

Reading: rest of 7.1, 7.2

What we did:

- Tuesday: We finished the proof of Theorem 7.6 and then proved Theorem 7.7 (i.e. that each K
_{λ}has a basis which is a disjoint union of cycles of generalized eigenvectors). We followed the arguments of the textbook (modulo some difference in notation). - Thursday: We concluded the proof of existence of Jordan canonical form. We then proved uniqueness of Jordan canonical form and discussed how to find a Jordan basis (section 7.2). The main result was the following:
Proposition: Let λ be an eigenvalue of T and γ

_{i}(1≤ i ≤ k) be disjoint cycles of generalized eigenvectors corresponding λ such that ∪ γ_{i}is a basis of K_{λ}. Then:- (a) The length of the longest cycle among the γ
_{i}equals the nilpotency index of (T-λI)_{Kλ}(which equals the smallest integer L such that dim(ker(T-λI)^{L}) equals the multiplicity of λ). - (b) For every r≥1, the number of cycles among the γ
_{i}of length at least r among the γ_{i}is equal to dim(Im(T-λI)^{r-1})-dim(Im(T-λI)^{r}). - (c) For every r≥1, the initial vectors of the cycles γ
_{i}of length at least r form a basis of Im(T-λI)^{r-1}∩E_{λ}. In particular, the number of cycles of length at least r equals the dimension of Im(T-λI)^{r-1}∩E_{λ}.

As a corollary of part (b), we saw that the Jordan canonical form of an operator is unique up to reordering the Jordan blocks (as if β=⊔

_{λ, i}γ_{λ,i}is a Jordan basis for T, with γ_{λ,i}a cycle of generalized eigenvectors corresponding to λ, then for each λ and r, the number of cycles γ_{λ,i}of length r (i.e. the number of Jordan blocks of size r with diagonal entry λ in the corresponding Jordan form of T) is independent of the Jordan basis).Parts (a) and (b) of the proposition help us find the Jordan canonical form: for each λ, part (a) helps us find the highest length, say L, of the cycles corresponding to λ, and then calculating the dimensions of Im(T-λI)

^{r}for r=L-1,...,0, and using part (b) of the proposition, we find the number of cycles of each length (and hence the dot diagram for λ).Finding a Jordan basis: Once we have found the Jordan canonical form, guided by part (c) of the proposition, we have the following general procedure for finding a basis of each K

_{λ}consisting of disjoint cycles. (The union of all these bases for various K_{λ}is a Jordan basis for T.)Draw the dot diagram for λ to see what is happening in each step. In what follows L is highest length of the cycles for λ.)

- Find a basis α
_{1}of Im(T-λI)^{L-1}∩E_{λ}. (These will be the initial vectors of cycles of length L in our basis of K_{λ}.) For each of the vectors v in α_{1}, form a cycle of length L with initial vector v. - Extend α
_{1}to a basis α_{2}of Im(T-λI)^{L-2}∩E_{λ}. (The vectors in α_{2}-α_{1}will be the initial vectors of the cycles of length L-1 in our basis.) For each vector v in α_{2}-α_{1}, form a cycle of length L-1 with initial vector v. - Continue in the same fashion. More precisely, in the r-th iteration, having already found a basis α
_{r-1}of Im(T-λI)^{L-r+1}∩E_{λ}, extend α_{r-1}to a basis α_{r}of Im(T-λI)^{L-r}∩E_{λ}. For each vector v in α_{r}-α_{r-1}, find a cycle of length L-r+1 with initial vector v. Stop after r=L. The cycles produced in this process are disjoint and linearly independent (by Theorem 7.6 of the textbook), and their union is a basis of K_{λ}(see the remark below).

Remark: The general procedure above is not in your textbook. We did not have time to explain why it works. Since the produced cycles are disjoint and their union is linearly independent, we just need to make sure that the total number of vectors in them equals the dimension of K

_{λ}. For this, we need to check that ∑_{r}r(dim(Im(T-λI)^{r-1}∩E_{λ})-dim(Im(T-λI)^{r}∩E_{λ}))=dim K_{λ}. This equation follows from part (c) of the proposition above, together with the fact that there exists a basis of K_{λ}which is a disjoint union of cycles of generalized eigenvectors for λ. Indeed, if ∪ γ_{i}with the γ_{i}disjoint cycles is such a basis for K_{λ}(note that we already know such basis exists), by part (c) of the proposition, the number of cycles of length r among the γ_{i}equals dim(Im(T-λI)^{r-1}∩E_{λ})-dim(Im(T-λI)^{r}∩E_{λ}), so that the total number of vectors in the cycles (i.e. dim(K_{λ})) equals ∑_{r}r(dim(Im(T-λI)^{r-1}∩E_{λ})-dim(Im(T-λI)^{r}∩E_{λ})). (In other words, stating the argument in the language of the dot diagram for λ, dim K_{λ}is the total number of dots, and r(dim(Im(T-λI)^{r-1}∩E_{λ})-dim(Im(T-λI)^{r}∩E_{λ})) is the total number of dots in the columns of length r.)We also briefly mentioned what we mean by a Jordan canonical form/basis in the context of matrices: by a Jordan canonical form/basis for A ∈ M

_{n×n}(F) over F we mean one for the map L_{A}:F^{n}→F^{n}. Equivalently, a Jordan canonical form for A over F is a Jordan matrix J such that J=P^{-1}AP for some P in M_{n× n}(F). (Columns of P form a Jordan basis.) By what we have proved in 7.1. and 7.2, if the characteristic polynomial of A splits over F, then A has a Jordan canonical form over F, which is unique up to reordering of the Jordan blocks. - (a) The length of the longest cycle among the γ

Next week's plan: Next week is the reading week. Classes and tutorials will resume in the week of Feb 25. Our midterm is on Friday March 1 (see the syllabus), and it covers the material covered up to now (5.1, 5.2, 5.4, 7.1, 7.2, and Appendix E).

Reading: Appendix E, some of 7.3

What we did: We went over the earlier assigned reading from Appendix E. We started the discussion on minimal polynomials (7.3).

Next week's plan: Finish 7.3 on Tuesday. Discuss rational canonical forms on Thursday.

Reading: rest of 7.3; 7.4

What we did: We finished 7.3 and started 7.4. We stated the main theorem (existence and uniqueness of rational canonical form), and looked at an example.

Next week's plan: We will give a sketch of the proof of existence of rational canonical form on Tuesday. (We won't discuss the proof of uniqueness in class. Problems 3 and 4 of Assignment 7 give the ingredients of the proof of uniqueness.) We will start chapter 6 on Thursday.

Reading: rest of 7.4, 6.1

What we did: On Tuesday we wrapped up 7.4 by giving a sketch of the proof of existence of rational canonical form. We started Chapter 6 on Thursday and finished 6.1 (almost, we didn't have time to define orthogonality).

Next week's plan: 6.2 and 6.3, maybe start 6.4.

Reading: 6.2

What we did: We did 6.2 on Tuesday and the first half of Thursday. We started 6.3 in the second half of Thursday lecture.

Next week's plan: 6.3 and 6.4

Reading: 6.3, 6.4, 6.6

What we did: We finished 6.3 on Tuesday. We did 6.4 and (essentially) 6.6 on Thursday.

Next week's plan: 6.5 and either 6.8 or another topic

Reading: 6.5, 6.8 (These two sections are not on the final exam.)

What we did: We spent Tuesday and the first half of Thursday on 6.5. We spent the second half of Thursday learning a bit about bilinear forms.