MAT247H1S Algebra II

Winter 2019

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.

Course Description

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)

Syllabus

Teaching Staff

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

Course Materials

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

midterm solutions (regular sitting)

Weekly Calendar

Week 1

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.

Week 2

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.

Week 3

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.
  We then started 5.4 by stating the Cayley-Hamilton theorem. We defined the notion of invariant subspaces and gave some examples. We showed that if W is a T-invariant subspace, then the characteristic polynomial of T_W (the restriction of T to W) divides the characteristic polynomial of T.

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.)

Week 4

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.

Week 5

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.
  Note: Our approach for this result was different from the textbook (where the statements (a) and (b) are proved assuming the characteristic polynomial splits). Here is a sketch/summary of how we proved (a): Let d=dim(K_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)^df(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).

• 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.

Week 6

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 α₁ 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 α₁, form a cycle of length L with initial vector v.
  • Extend α₁ to a basis α₂ of Im(T-λI)^L-2∩E_λ. (The vectors in α₂-α₁ will be the initial vectors of the cycles of length L-1 in our basis.) For each vector v in α₂-α₁, 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:Fⁿ→Fⁿ. Equivalently, a Jordan canonical form for A over F is a Jordan matrix J such that J=P^-1AP 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.

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).

Week 7

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.

Week 8

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.

Week 9

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.

Week 10

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

Week 11

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

Week 12

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.