Math 2135 (Fall 2020):
Linear Algebra for Math Majors

[course policies] [lecture notes] [lecture video] [feedback]

Lectures

All lectures are now on Zoom in Meeting Room 576-664-005.
  1. April 24: [video]
  2. April 22: [video]
  3. April 20: [video]
  4. April 15: [video]
  5. April 10: [video]
  6. April 8: [video]
  7. April 1: [video]
  8. March 30: [video]
  9. March 16: [video]

Handouts

Homework

  1. For Wednesday, April 29: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail or contact me for the meeting password), at the usual scheduled time (9am). In the last class, I will do my best to answer your questions on any subject (religion and politics excepted).
  2. For Monday, April 27: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail or contact me for the meeting password), at the usual scheduled time (9am). Do LADW, Chapter 4, §2, Exercises 2.8, 2.10, 2.12, 2.13.

    We proved in class that, if \( T : V \to V \) is a linear transformation then there is a basis \( X \) of \( V \) such that \( [T]^X_X \) is upper triangular. Show that the eigenvalues of \( T \) are the diagonal entries of \( [T]^X_X \). Conclude that \( \det(T) \) is the product of the eigenvalues of \( T \) and that \( \operatorname{trace}(T) \) is the sum of the eigenvalues of \( T \).

    Write a formula for the characteristic polynomial \( p(\lambda) \) of the matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). What do you get if you evaluate \( p(A) \)?
  3. For Friday, April 24: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail or contact me for the meeting password), at the usual scheduled time (9am). Please read LADR, 5.22–5.27. Do LADW, Chapter 1, §3, Exercise 3.6. Prove that if \( V \) is a complex vector space with a basis \( \mathbf x^1, \ldots, \mathbf x^n \) then the vectors \( \mathbf x^1, i \mathbf x^1, \mathbf x^2, i \mathbf x^2, \ldots, \mathbf x^n, i \mathbf x^n \) are a basis of \( V \) as a real vector space. Do LADR, Exercises 5.B, #4.
  4. For Wednesday, April 22: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail or contact me for the meeting password), at the usual scheduled time (9am). Please read LADR, pp. 21–24 and LADW, §§2.4–2.7. What LADW called "linearly independent subspaces" are called "direct sums" in LADR; direct sum is the more standard term. Do LADW, Chapter 4, §2, Exercises 2.1 and 2.3.
  5. For Monday, April 20: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Please read LADW, Chapter 4 through §2.3 and do LADW, Chapter 4, §1, Exercises 1.1–1.6.
  6. For Friday, April 17: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Please watch this video about complex numbers and the fundamental theorem of algebra. You may also find the review of complex numbers on Khan Academy helpful.
  7. For Wednesday, April 15: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Please read LADW, Chapter 4, §§1.5–1.7 and complete Handout #8.
  8. For Monday, April 13: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Read LADW, Chapter 4, §§1.1–1.4. Watch EoLA, Chapter 14. In class, we will work on Handout #8.
  9. For Friday, April 10: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Do LADW, Chapter 3, §7, Exercises 7.1–7.5.
  10. For Wednesday, April 8: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Please read LADW, Chapter 3, §3.5. Do LADW, Chapter 3, §3, Exercises 3.6, 3.8, 3.10–3.12.
  11. For Monday, April 6: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). We will discuss Handout 7.
  12. For Friday, April 3: Class will be on Zoom in Meeting Room 576-664-005 (check your e-mail for the meeting password), at the usual scheduled time (9am). Please read LADW, Chapter 3, §4 and do LADW, Chapter 3, §3, Exercises 3.1–3.3, 3.5, 3.7, 3.9. Note: we will have a lecture instead of a worksheet to finish our introduction to the determinant.
  13. For Wednesday, April 1: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). Please watch EoLA, Chapter 6 and read LADW, §§3.1–3.3. Prove that you can transform any invertible square matrix into a diagonal matrix (every entry that is not on the main diagonal is 0) using only the row operation that adds a multiple of one row to another.
  14. For Monday, March 30: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). We will begin discussing determinants.
  15. For Friday, March 20: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). Do LADW, Chapter 2, §7, Exercises 7.2, 7.4, 7.8. In class we will discuss Handout 6.
  16. For Wednesday, March 18: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). Read LADW, Chapter 2, §7. Do LADW, Chapter 2, §6, Exercise 6.2 and LADW, Chapter 2, §7, Exercises 7.1, 7.3.
  17. For Monday, March 16: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). Please create an account on Overleaf in order to collaborate during our meeting. If you prefer, you may also be able to collaborate effectively using the desktop version of Microsoft Word which has a powerful equation editor that supports LaTeX. (I have not yet tested this.)

    In class we will discuss Handout 5 [overleaf]. This is a slightly modified version of Handout 4.
  18. For Friday, March 13: Class will be on Zoom in Meeting Room 576-664-005, at the usual scheduled time (9am). There will be no meeting in person. Download Zoom here. We will discuss the problems in Handout #4 [overleaf]. Please make sure you have an account at Overleaf before class.

    In class, we will discuss Handout #4, Problems 2 and 3.
  19. Read LADW, Chapter 2, §6. Do LADW, Chapter 2, §4, Exercise 4.1. Let \( A = \begin{pmatrix} 1 & 3 \\ -2 & 1 \\ 0 & -2 \end{pmatrix} \). Find a matrix \( B \) such that \( \operatorname{col}(A) = \operatorname{null}(B) \). Do LADW, Chapter 2, §5, Exercises 5.1, 5.4, 5.6.
  20. For Wednesday, March 11: Read LADW, Chapter 2, §5. There are no ungraded problems for Wednesday. The graded problem set is due on Tuesday, March 10.
  21. For Monday, March 9: Read LADW, Chapter 2, §4. Do LADW, Chapter 2, §3, Exercises 3.5, 3.6. There is a graded problem set due on Tuesday, March 10.
  22. For Friday, March 6: Read LADW, Chapter 2, §3. Prove the following theorem: If \( U \xrightarrow{T} V \xrightarrow{S} W \) are linear transformations then \( \operatorname{im}(S \circ T) \subset \operatorname{im}(S) \). If \( T \) has a right inverse then \( operatorname{im}(S \circ T) = \operatorname{im}(S) \). Use this theorem to show that if \( A \) is a matrix then its column space is unaffected by elementary column operations. Do LADW, Chapter 2, §2, #2.1(c) and LADW, Chapter 2, §3, #3.2.
  23. For Wednesday, March 4: Read LADW, Chapter 2, §3. Watch EoLA, Chapter 7. Do LADW, Chapter 2, §2, Exercise 2.1(b). Do LADR, §3.B, Exercises #2.
  24. For Monday, March 2: Read LADR, §§3.25–3.28. Do LADW, Chapter 2, §2, Exercises 2.1(a), 2.2 (in 2.2, express the set of solutions as the kernel of a linear transformation). Do LADR, §3.B, #3, 5, 9.
  25. For Friday, February 28: Read LADR, §§3.12–3.21 and LADW, Chapter 2, §2. Do LADW, Chapter 1, §7, Exercises 7.2–7.4.

    Let \( P_3 \) be the vector space of polynomials of degree \( \leq 3 \). Let \( T : P_3 \to \mathbf R^2 \) be the linear transformation \( T(f) = \begin{pmatrix} f(2) \\ f(-3) \end{pmatrix} \). Describe the kernel of \( T \) in more concrete terms. Compute a basis for the kernel of \( T \) (Hint: there is a way to do this with minimal calculation).
  26. For Wednesday, February 26: Read LADR, §1.32–1.39. Do LADW, Chapter 1, §7, Exercises 7.1 and LADR, §1.C, Exercises 1, 3, 9.
  27. For Monday, February 24: The second graded problem set is due on Saturday. Read LADW, Chapter 1, §7. Complete the handout started in class on Friday.
  28. For Friday, February 21: Read LADR, pp. 82–83 (ignore the discussion of dimension for now). Do LADW, Chapter 2, §6, Exercise 6.10 and LADW, Chapter 2, §8, Exercises 8.1, 8.2, 8.3.

    Let \( P_2 \) be the vector space of polynomials of degree \( \leq 2 \) with real coefficients. Define \( T : P_2 \to \mathbb R^2 \) by the formula \( T(f) = \begin{pmatrix} f(0) \\ f(1) \\ f(2) \end{pmatrix} \). Find a formula for the inverse linear transformation \( T^{-1} : \mathbb R^2 \to P_2 \).
  29. For Wednesday, February 19: Revisions for Exam #1 are due. Do LADW, Chapter 1, §6, Exercises 6.1, 6.5–6.7, 6.9, 6.11. Hint for 6.11: Find a basis of \( \mathbb R^3 \) where the matrix is easy to describe and then change basis to the standard basis.
  30. For Monday, February 17: Read LADW, Chapter 1, §6 and LADW, Chapter 2, §8. Do LADW, Chapter 1, Exercise 5.8 and LADW, Chapter 1, Exercises 6.3, 6.4. Let \( V \) be the vector space of straight-line motions in the plane and let \( T_\theta : V \to V \) be the linear transformation that rotates vectors counterclockwise by the angle \( \theta \). Show that \( T_\theta \) is invertible and use the matrix of the inverse to show that \( \cos(\theta) = \cos(-\theta) \) and \( \sin(-\theta) = - \sin(\theta) \).
  31. For Friday, February 14: Make sure you are caught up with all of the reading from before the exam. Do LADW, Chapter 1, §5, Exercise 5.1 and LADR, Exercise 3.A.6x
    We have proved in class that, if \( T : U \to V \) is a linear transformation between vector spaces \( U \) and \( V \) with bases \( X = \begin{pmatrix} \mathbf x^1 & \cdots & \mathbf x^n \end{pmatrix} \) and \( Y \), respectively, then, for any vector \( \mathbf u \) in \( U \), we have the formula \( [T]^X_Y [\mathbf v]_X = [T(\mathbf v)]_Y \). Use this formula to prove that if \( S : V \to W \) is an additional linear transformation and \( Z \) is a basis of \( W \) then \( [S \circ T]^X_Z = [S]^Y_Z [T]^X_Y \).
  32. For Wednesday, February 12: Read LADW, Chapter 1, §5. Do LADW, Chapter 1, §3, Exercise 3.5. Read LADR, §3.A. Do LADR, Exercise 3.A.4.. Watch EoLA, Chapter 4.
  33. For Monday, February 10: Prepare for the exam.
  34. For Friday, February 7: The exam is postponed until Monday, February 10.
  35. For Wednesday, February 5: There is an in-class exam on Friday, February 7.

    Read LADW, Chapter 1, §4. Do LADW, Chapter 1, §3, Exercises 3.2–3.4. When LADW asks you to find the matrix of a transformation without specifying a basis of \( \mathbb R^n \), it wants you to use the basis \( E = \begin{pmatrix} \mathbf e^1 & \cdots & \mathbf e^n \end{pmatrix} \), where \( \mathbf e^i \) is the column vector with a \( 1 \) in the \( i \)th position and \( 0 \) in all other positions.

    Suppose that \( V \) is a vector space with basis \( X = \begin{pmatrix} \mathbf x^1 & \cdots & \mathbf x^n \end{pmatrix} \). Show that the transformation \( T : V \to \mathbb R^n \) given by the formula \( T(\mathbf v) = [\mathbf v]_X \) is a linear transformation.
  36. For Monday, February 3: Read LADW, Chapter 1, §3. Watch EoLA, Chapter 3. Do LADR, Exercises 2.A.8, 2.A.9 and 3.A.1

    Let \( V \) be the vector space of straight-line motions in the plane and let \( \mathbf x^1 \) be any nonzero vector in \( V \). Let \( \mathbf x^2 \) be the vector that is 90° (\( \pi / 2 \) radians) counterclockwise of \( x^1 \), with the same length. Let \( \theta \) be any real number and let \( T : V \to V \) be the linear transformation that rotates vectors counterclockwise by the angle \( \theta \). Compute \( [T(\mathbf x^1)]_X \) and \( [T(\mathbf x^2)]_X \), where \( X \) is the basis \( \begin{pmatrix} \mathbf x^1 & \mathbf x^2 \end{pmatrix} \). Use this calculation to write a formula for \( T(a \mathbf x^1 + b \mathbf x^2) \) for all \( a , b \in \mathbb R \).
  37. For Friday, January 31: Read LADW, §3–3.1 (pp. 12–13). Do LADW, Chapter 1, Exercises 2.3–2.6 and LADR, Exercises 2.A.1–2.A.3, 2.A.6 and 2.B.3(ab).
  38. For Wednesday, January 29: Read LADR, §2.B (skip 2.34). Do LADW, Section 2, Exercises 2.3, 2.5 and LADR Exercise 2.B.6.

    The vectors \( \mathbf x^1 = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \) and \( \mathbf x^2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \) form a basis of \( \mathbb R^2 \). Find the coordinates of the vectors \( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) in the basis \( X = \begin{pmatrix} \mathbf x^1 & \mathbf x^2 \end{pmatrix} \). Find a general formula in terms of \( a \) and \( b \) for the coordinates of \( \begin{pmatrix} a \\ b \end{pmatrix} \) in the basis \( X \).
  39. For Monday, January 27: Watch ELA, Chapter 2 (this was already assigned for Friday, but posted late, so please watch if you haven't already). Read LADR, pp. 32 – 34 and LADW, §8.1 (this is less than a page).

    Do LADW, Chapter 1, Exercises 2.1 – 2.

    In the following questions, let \( V \) be the vector space of straight-line motions in the plane and let \( W \) be the vector space of straight-line motions in 3-dimensional space.
    1. Draw two vectors in \( V \) that are linearly dependent.
    2. Can you find two vectors in \( V \) that are linearly dependent and generate \( V \)?
    3. How many vectors are necessary to generate \( V \)?
    4. How big is the largest set of linearly independent vectors in \( V \)?
    5. How many vectors are necessary to generate \( W \)?
    6. How big is the largest set of linearly independent vectors in \( W \)?
  40. For Friday, January 24: The first graded problem set is due. Read LADW, §1.2 and LADR, §§2.3–2.6, 2.8–2.15 and watch ELA, Chapter 2. Do LADR, Exercises 1.B.1–4 (please prioritize the graded homework assignment over these exercises).
  41. For Wednesday, January 22: Read LADR, §1.B. Do LADW, Chapter 1, Exercises 1.2 – 1.7. (You may notice that solutions to some of these exercises are given in the reading assignment in LADR. Find a way to make the most of having these solutions, for example by referencing them only after trying the problems yourself, or reading the solution of only one of them.) Upload a PDF as test submission on canvas.
  42. For Friday, January 17: Read LADW, Ch. 1, §1. Do Exercises 1.3 and 3.1 in Chapter 1 of LADW.
  43. For Wednesday, January 15: Read the course policies. There will be an ungraded quiz. Make sure you can access Linear algebra done wrong and Linear algebra done right (the latter should be available through university subscriptions; you may have to download it while on campus). Read the first example in Linear algebra done wrong, §1.1 and do Exercise 1.1. Watch Essence of linear algebra, chapter 1.

Graded assignments and exams

Rules for collaboration on problem sets: You may use any textbooks or video resources you want. You may discuss the problems verbally with others (either in the class or not) without writing anything down (this includes typing) and without making any notes or recordings. When writing your solutions, you should work completely independently: no discussion at all is allowed during the writing process.

Rules for submission of problems sets: Problem sets should be submitted in PDF format, as a single file (do not submit each page as an individual file). Legibility is important. Typed solutions are appreciated but not required (you can use the tex or overleaf links below to get a LaTeX template to start typing your solutions).

  1. Due Wednesday, May 6: Problem Set #5 [tex] [overleaf] [canvas] [solutions]
  2. Due Friday, April 17: Problem Set #4 [tex] [overleaf] [canvas] [solutions]
    Special notes: submissions in groups of up to 2 students are allowed (please contact me if you need help finding a partner); I am happy to be flexible with the due date if your circumstances require it.
  3. Due Tuesday, March 10: Problem Set #3 [tex] [overleaf] [canvas]
  4. Due Saturday, February 22: Problem Set #2 [tex] [overleaf] [canvas] [solutions]
  5. Monday, February 10: Exam #1. Topics: linear independence, spanning, basis
  6. Due Friday, January 24: Problem Set #1 [tex] [overleaf] [canvas] [solutions]