# Math V2020 - Honors Linear Algebra

Much of this page was shamelessly copied from Robert Lipschitz's page from when he taught the course in the Fall of 2008.

Basic information:

1. Time: TTh 2:40 -- 3:55 PM
2. Place: Math 407
3. Textbook: Linear Algebra by Klaus Jänich
4. Office hours: Monday, 10:00-11:00 AM, 1:00-2:00 PM in Math, Rm 523
5. Teaching assistant: Yifei Zhao. His help room hours are Mondays, 2-4pm, in Room 406, Mathematics
6. Help room hours
7. Final exam: TBA

## Description and goals.

At heart, linear algebra is about linear equations and linear transformations. Linear algebra's importance to both mathematics and its applications rivals -- and perhaps exceeds -- that of calculus. Unlike calculus, however, linear algebra becomes clearer in a somewhat more abstract setting -- that of vector spaces, linear transformations and inner products. This course will discuss abstract linear algebra. Two things will keep us anchored: a multitude of concrete examples, computations and applications; and proofs, to track what is true in general -- and why.

The main goals of the course are:

• To provide the essential tools from linear algebra needed in mathematics, science and engineering.
• To provide an introduction to writing and reading mathematics, suitable for taking more advanced courses or studying further mathematics on one's own.
• To begin to explore a deep and beautiful subject, and its relationships with other parts of mathematics and science.

## How is V2020 different from V2010?

While there will be a lot of overlap between "Linear Algebra" and "Honors Linear Algebra," there will also be several differences. In particular, V2020 will:

• Present somewhat more material than V2010. In particular, we will talk about Jordan Normal Form and several versions of the Spectral Theorem.
• Work in a more abstract setting -- that of vector spaces and linear maps -- than V2010. This requires a change in mind set at the beginning, which can be difficult. Ultimately, however, the ideas are clearer from a more abstract perspective.
• Involve writing proofs. By the end of the semester students should be able to write clear mathematical arguments. (Students are not expected to have had experience writing proofs before taking the course.)

Consequently, the material in V2020 will be presented at a somewhat higher pace than in V2010. Although students in V2020 will learn to compute everything that students in V2010 do, there will be somewhat less practice computing. We will also talk about some applications of linear algebra but may not spend as much time on applications as V2010.

You should strongly consider V2020 instead of V2010 if:

• You are planning to major in mathematics.
• You enjoy thinking about (abstract?) mathematical questions.
• You would like a sense of what more advanced mathematics courses are like.
• You are planning to take more advanced mathematics courses later.
• You want a deeper background in mathematics for your major or other studies.

You should not take V2020 if:

• You already took V2010 or V1207-1208.
• You do not like thinking about abstract mathematics (or strongly prefer numbers to variables, say).
• You are only taking linear algebra because of a degree requirement.

## Policies

The grade will be determined by a weighted average of the homework scores, the midterm, and the final exam. Roughly the percentages will be 30%, 30%, 40%.

### Homework

There will be weekly problem sets due each Tuesday one week after they appear on the webpage here (below). To hand in homework, please find the dropbox marked V2020 on the fourth floor of the math building. You can hand in late, but every day late will cost you 10% of the score on that set. Late means after 6PM.

### Missed exams

If you have a conflict with any of the exam dates, you must contact me ahead of time so we can make arrangements. If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can.

## Syllabus and schedule

This will change as we go along. Namely, the pace set by the schedule below is too optimistic and we will move things later.

Date Material Textbook Announcements
09/02 sets, maps 1.1, 1.2
09/04 vector spaces, complex numbers, subspaces 2.1, 2.2, 2.3
09/09 fields, independence, span 2.5, 3.1 Problem set 1 due
09/11 bases, dimension 3.1, 3.2
09/16 dimension (proofs), linear maps 3.4, 4.1 Problem set 2 due
09/18 linear maps, matrices 4.1, 4.2
09/23 matrices, rotations, multiplication 4.2, 4.5, 5.1 Problem set 3 due
09/25 multiplication, rank 5.1, 5.2
09/30 elementary, inverting, Gaussian 5.3, 5.5, 7.3 Problem set 4 due
10/02 systems, Gaussian 7.1, 7.3
10/07 Cramer's rule and determinants 6.1, 6.2, 7.1 Problem set 5 due
10/09 determinants 6.3, 6.4, 6.5, 6.7
10/14 Review 1.1 - 7.5 Problem set 6 due
10/16 Midterm on material above 1.1 - 7.5
10/21 Inner products, orthogonal vectors 8.1, 8.2 Problem set 7 due
10/23 Orthogonal maps, groups 8.3, 8.4
10/28 Symmetric Bilinear forms, completing the square, subgroups, Orthogonal groups Ch 8 Problem set 8 due
10/30 Eigenvalues, Characteristic Polynomial 9.1, 9.2
11/4 Election day ---
11/6 Polynomials 9.4 Problem set 9 due
11/11 Slef-adjoint, Symmetric 10.1, 10.2 Problem set 10 due
11/13 Principal axes 10.3
11/18 Rank Thm, Jordan Normal Form, Nilpotent Endos 11.2, 11.3, Extra Problem set 11 due
11/20 Jordan Normal Form 11.3, Extra
11/25 Examples of JNF Problem set 12 due
11/27 Thanksgiving ---
12/2 Sylvester Inertia 11.5 Problem set 13 due
12/4 Review
12/18 EXAM 1:10-4:00 Rm 407 All sections of the book listed above + extra material listed below

## Extra theory

Some of the material we discussed in the lectures is not in the book. Here is a list of these things:

1. Row echelon form. See for example Wikipedia here
2. Elementary matrices and their relationship with row operations. See for example Wikipedia here
3. Symmetric bilinear forms and symmetric matrices. See the two sections on Definition and Matrix representation in wikipedia.
4. Subgroups. See the first part of wikipedia).
5. If V is a Euclidean vector space and f : V ---> V is an endomorphism of the vector space V, then f is self-adjoint if and only if the map Bf : V x V ---> R, (v, w) |---> <f(v), w> is a symmetric bilinear form.
6. If V is a finite dimensional Euclidean vector space and B : V x V ---> R is a symmetric bilinear form, then there is a unique self-adjoint endomorphism f : V ---> V of V such that B = Bf.
7. Nilpotent endomorphisms. See Lecture 21. There is a typo on page 149: the statement is missing "(v_1, ..., v_n) is in Jordan Normal Form". There is a mistake on page 153: the upper left corner of the displayed matrix is wrong unless we use v_m, ..., v_1, w_1, ..., w_{dim(W)} as our basis.
8. Proof of Jordan Normal Form. See Lecture 22. Observe how one of the pages is a page from my own research; it is the unique page without a page number, please skip it.

## Problem sets

The book has this interesting feature where in each chapter there is a test. When I ask below that you do these, it means that you go through the test, recording your answers to the multiple choice questions, and then you check in the back of the book as to whether you got them right. If not, please look at the suggestions given below the answers as to how to improve. Do not hand in the tests.

1. Problem set 1
1. Do test 1.3 from the book.
2. From Section 1.5 do exercises 1.1, 1.2 (prove the displayed statement), and 1.3.
3. Do test 2.4 from the book.
4. From Section 2.9 do exercises 2.1 (list axioms you are using in each step as explained in the book), 2.2, and 2.3.
2. Problem set 2
1. Prove, using the axioms, that in a field the product of two nonzero elements is always nonzero.
2. From Section 3.7 do exercises 3.1, 3.2, and 3.3.
3. We will use R to denote the field of real numbers and we will denote Rn to denote the usual vector space of n-tuples of real numbers.
1. Find a basis for V = {(x,y) in R2 | 3x + 5y = 0}. What is the dimension of V?
2. Find a basis for W = {(x,y,z) in R3 | x + 2y + 4z = 0}. What is the dimension of W?
3. The vector (10, -1, -2) is an element of W. Write it as a linear combination of the basis vectors you found above.
4. Let P be the vector space of all polynomials in x over the real numbers. Prove that P is infinite dimensional.
3. Problem set 3
1. Do test 3.3 from the book.
2. From Section 4.7 do exercise 4.1.
3. Let F : R2 ---> R2 be the linear map given by F(x, y) = (3x + 2y, -6x -4y).
1. Find a basis for the kernel of F.
2. Find a basis for the image of F.
3. Draw the kernel of F and the set {(x, y) such that F(x, y) = (3, -6)} in the same picture.
4. How many solutions are there to the equation F(x, y) = (1, 1)?
4. Let F : R3 ---> R3 be the linear map given by F(x, y, z) = (2x + y + 8z, y + 2z, x + y + 5z).
1. Find a basis for the kernel of F.
2. Find a basis for the image of F.
3. Find a vector v in R3 which is not in the image of F.
4. Find all the solutions to the equation F(x, y, z) = (2, 0, 1).
5. Give an example of a surjective linear map R3 ---> R3. (Yes, this is silly.)
6. Let P be the real vector space of all real polynomials which you have previously shown to be infinite dimensional. This exercise shows you how infinite dimensional vector spaces behave differently from finite dimensional ones.
1. Give an example of an injective but not surjective linear map f : P ---> P.
2. Give an example of a surjective but not injective linear map g : P ---> P.
4. Problem set 4
1. Do test 4.3 from the book.
2. From Section 4.7 do exercise 4.2.
3. Compute the following matrix multiplications (sorry about the type setting, I do not know how to make round braces using html)
1. The following product
 1 0 -1 0 2 4

 1 0 -1 2 1 2
2. The following product
 1 0 -1 2 1 2

 1 0 -1 0 2 4
3. Compute the 8th power of the matrix   A =
 1 2 2 3
in other words, compute the 8-fold self-product A A A A A A A A. (Hint: to do this you only need to do 3 matrix multiplications.)
4. Let   A =
 1 4 2 9
. Find the inverse B of A, in other words, the matrix whose associated linear map is the inverse of the linear map associated to A. Here I want you to use the following approach: write B =
 a b c d
The condition that B is the inverse of A boils down to a system of linear equations which you can solve.
5. Multiplication with elementary matrices. No explanations necessary.
1. Let E =
 1 0 0 0 17 0 0 0 1
and let A be a 3x3 matrix. Describe in your own words: What is the effect of left multiplying A by E? What is the effect of right multiplying A by E?
2. Let E =
 1 0 17 0 1 0 0 0 1
and let A be a 3x3 matrix. Describe in your own words: What is the effect of left multiplying A by E? What is the effect of right multiplying A by E?
6. In this problem we let f : R2 ---> R2 be the linear map given by f(x, y) = (3x + 4y, -x). Let v_1 = (1, 1), v_2 = (1, 0) and let w_1 = (2, 1) and w_2 = (0, 2). What is the matrix A of f with respect to the bases (v_1, v_2) and (w_1, w_2)? [In other words, your matrix A = (aij) should have the property that f(v_j) = a1j w_1 + a2j w_2 for j = 1, 2.]
5. Problem set 5
1. Do the test 5.4 from the book.
3. Read about elementary matrices and their relationship with row operations, for example on Wikipedia here
4. Using elementary row operations (one per step) reduce each of the following matrices to row echelon form:
 1 2 3 2 3 4 3 4 5
and
 1 2 3 4 1 -2 -3 -4 1 2 -3 -4 3 2 -3 -4
and
 1 1 1 1 1 1 1 1 100
5. Use the process described in class and in Section 5.5 of the book to invert the matrix   A =
 5 3 3 2
6. Use the process described in class and in Section 5.5 of the book to invert the matrix
 0 1 1 1 0 1 1 1 0
7. Use Gaussian elimination (row reduction on extended matrix) to find all solutions in R3 to the following system of equations 2x + 3y + z = 0, x + y + z = 0, 3x + 4y + 2z = 0, y + z = 0.
8. Use Gaussian elimination (row reduction on extended matrix) to find all solutions in R3 to the following system of equations 3x + y - 3z = 14, 2x + y - 3z = 9, -2x - y + 4z = -8.
9. Use Gaussian elimination (row reduction on extended matrix) to find all solutions in R3 to the following system of equations x + y + 3z = 5, -2x - 2y - 6z = -20.
6. Problem set 6
1. Do test 6.6 from the book.
2. Compute the determinant of the matrix
 1 2 3 4 5 6 -1 -2 -1
and please show explicitly how you did the computation.
3. Compute the determinant of the matrix
 0 5 0 0 0 12 11 2 0 10 3 13 0 0 14 9 8 6 1 7 0 15 0 0 4
and please show explicitly how you did the computation.
4. Compute the determinant of the matrix
 a b 0 0 0 0 c d 0 0 0 0 0 0 e f 0 0 0 0 g h 0 0 0 0 0 0 i j 0 0 0 0 k l
and please explain how you did the computation.
5. Let A be an n-by-n matrix. Let λ be a scalar. Prove that det(λ A) = λn det(A).
6. Let A be the n-by-n matrix which has a 1 in the upper right hand corner and 1's just below the diagonal and 0's everywhere else. For example, if n = 4 then you get
 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0
. Find a formula for the determinant of A in terms of n and prove it carefully.
7. Suppose that A is a n-by-n matrix all of whose entries are even integers. Prove that det(A) is an even integer.
7. Problem set 7
1. Do test 7.4 from the book (I should have asked you to do this last week).
8. Problem set 8
1. Do test 8.5 from the book.
2. With the standard inner product on R3 compute the lengths of (1, 1, 1) and (1, 2, 3) and the angle between them. The answer is not "nice" -- it is fine if you explain exactly what the final calculation step is without actually putting it into a calculator.
3. From Section 8.7 do exercises 8.1, 8.2, 8.3.
4. Use the Gram-Schmidt process to find an orthnormal bases for the image of the matrix
 2 1 1 1 1 2
with respect to the standard inner product.
5. Let V = R2. Show that the rule defined by = 2x_1y_1 + 3x_1y_2 + 3x_2y_1 + 5x_2y_2 defines an inner product on V.
9. Problem set 9
1. Read up about symmetric bilinear forms and symmetric matrices (use your notes, or the two sections on Definition and Matrix representation in wikipedia).
2. For which values of t in R does the symmetric matrix
 t 2 1 2 t 1 1 1 t
define a positive definite symmetric bilinear form. Be precise and explain your argument.
4. Consider the group R*=R - {0} of nonzero real numbers with group law given by multiplication. Find a subgroup with two elements and prove there is no subgroup with 3 elements.
5. Show that if A is an nxn matrix with integer entries which is invertible, then the inverse has integer entries also if and only if the determinant of A is 1 or -1. (Hint: In one direction use the determinental formula for the inverse. For the other direction use that the determinant is multiplicative.)
6. Use the result of the previous exercise to prove GL(n, Z) = {A in M(n x n, Z) with det(A) = 1 or -1} is a subgroup of GL(n, R) but that on the other hand {A in M(n x n, Z) with det(A) not zero} is not a subgroup of GL(n, R).
7. Let V = {(x_1, x_2, x_3, ...) with x_n in C} the complex vector space of infinite sequences of complex numbers. Addition and scalar multiplication is componentwise, as in the case of the standard vector space Cn. Let f : V ---> V be the endomorphism which sends (x_1, x_2, x_3, ...) to (0, x_1, x_2, x_3, ...). Show that f has no eigenvector.
8. Compute the eigenvalues and eigenvectors of the matrix
 1 10 100 0 2 20 0 0 3
Hint: First guess the eigenvalues and then compute the eigenvectors. You may use theory from the book about how many eigenvalues there can be even if we haven't covered it yet in the lectures.
10. Problem set 10
1. Do test 9.3 from the book.
2. Do exercise 9.1 from Section 9.5 of the book.
3. Let n, m be positive integers. Let A, B be square matrices of size n and m. Let C be an n x m matrix. Consider the square matrix of size n + m which in block form looks like this:
 A C 0 B
. What this means is that the entries in the left-lower m x n block are all zero. Show that the determinant of this matrix is equal to the product of det(A) and det(B).
4. Find the roots and the algebraic multiplicities of the polynomial x^7 - x^6 - 6*x^5 + 10*x^4 + 5*x^3 - 21*x^2 + 16*x - 4 over the complex numbers. Totally fine if you use some computer algebra to do this. Just the answer is OK here.
5. Find the roots and the algebraic multiplicities of the polynomial x^4 + 3*x^2 + 1 over the complex numbers. Same remarks as previous question.
11. Problem set 11
1. Do test 10.4 from the book.
2. Find a real 3 x 3 matrix which has eigenvalue 11 with algebraic and geometric multiplicity 1 and no other eigenvalues. Please explain why the example works.
3. Find a real 6 x 6 matrix A which has eigenvalue 13 with algebraic multiplicity 4 and geometric multiplicity 2 and eigenvalue 17 with algebraic and geometric multiplicty 2.
4. From Section 10.5 of the book do exercise 10.1.
5. Give an example of a 2 x 2 complex symmetrix matrix A which is not diagonalizable over the complex numbers.
6. Call a complex n x n matrix A Hermitian if its transpose is equal to its complex conjugate: in other words the entry aij is equal to the complex conjugate of aji. Recall that to get the complex conjugate of a complex number, you change the sign of the imaginary part. In particular, a real symmetrix matrix is Hermitian when viewed as a complex matrix. Show that a nonzero 2 x 2 Hermitian matrix A has two distinct real eigenvalues (in particular it is diagonalizable over the complex numbers). [Edit 11/24/2014: This is not quite correct, it should say that the complex eigenvalues of A are real and that A is diagonalizable over the complex numbers.]
12. Problem set 12.
1. Let f : V ---> V be an endomorphism of a vector space. Let v be a vector and m > 0 an integer such that fm(v) = 0 but fm - 1(v) is not zero. Prove that v, f(v), f2(v), ..., fm - 1(v) are linearly independent.
2. Let U, W be subspaces of a vector space V such that U ∩ W = 0. Let v be a vector in V which is not in U + W. Let W' be the span of W and v, in other words W' = {w + λ v | w ∈ W, λ is a scalar}. Show that U ∩ W' = 0.
3. Do exercise 11.2 from Section 11.7 of the book.
4. For every triple a, b, c of complex numbers determine the Jordan normal form for the matrix
 0 0 abc 1 0 -ab - ac - bc 0 1 a + b + c
Hint: Compute the characteristic polynomial and find its roots in terms of a, b, c. Then carefully distinguish the cases where some of the roots coincide.
5. Suppose a linear self map f : V ---> V of a finite dimensional complex vector space has eigenvalues λi where i = 1, ..., r. What are the eigenvalues of f^n?
6. Is the result you stated (and argued) in the previous execise also true for an endomorphism of a finite dimensional real vector space?
13. Problem set 13: Empty problem set.