**Instructor:** Fedor Manin

**Sections** 0020 and 0075, Spring 2018

**Lecture**: 9:10AM MWF (section 0020), 12:40PM MWF (section 0075)

**Office**: Math Building 318 (not Math Tower 318 or Cockins Hall 318)

**Office hours**: 10:15–11:30AM Monday and Friday and 1:45–3:00PM Wednesday, or by appointment, or at additional times to be announced on Carmen.

**Textbook**: *Introduction to Linear Algebra* by Johnson, Riess, and Arnold

**Grader**: Yilong Wang

Click here for the syllabus and
**tentative** semester schedule. See also
Carmen.

**Homework 1**should be done by Wednesday, January 17 to prepare for the quiz on that day. Here are the textbook pages you need.**Homework 2**should be done by Friday, January 26 to prepare for the quiz on that day. Here are the textbook pages you need.**Homework 3**should be done by Friday, February 2 to prepare for the quiz on that day.**Homework 4**should be done by Friday, February 9 to prepare for Midterm I. I’ve also made a**list of topics**for Midterm I. Let me know if you think something is missing from it. I'll either add it in or tell you not to worry about it.**Homework 5**should be done by Friday, February 16 to prepare for the quiz on that day.**Homework 6**should be done by Friday, February 23 to prepare for the quiz on that day.**Homework 7**should be done by Friday, March 2 to prepare for the quiz on that day.**Homework 8**should be done by Friday, March 9 to prepare for the quiz on that day.**Homework 9**should be done by Wednesday, March 21 to prepare for the quiz on that day.**Homework 10**(answer key here) should be done by Wednesday, March 28 to prepare for Midterm II. I’ve also made a**list of topics**and a set of**selected homework solutions**for Midterm II.**Homework 11**should be done by Wednesday, April 4 to prepare for the quiz on that day.**Homework 12**should be done by Wednesday, April 11 to prepare for the quiz on that day.**Homework 13**should be done by Wednesday, April 18 to prepare for the quiz on that day.**Homework 14**covers material that was not on the previous homeworks—roughly §4.7. I’ve also made a**list of topics not covered on the midterms**. Both are subject to change, for now; let me know if you have any questions. An answer key for Homework 14 will be posted over the weekend, but please spend time thinking about the problems before you look at it.

**Jan. 8th**: Introduction. What is a function? Linear equations.**Jan. 10th**: §1.1: Systems of linear equations. Geometry of solution sets in 2 and 3 dimensions. Every system of linear equations has zero, one, or infinitely many solutions. Elementary operations.**Jan. 12th**: §1.2: Gauss–Jordan elimination. Interpreting a matrix in reduced echelon form as a solution set to a system of equations.**Jan. 17th**: §1.3: More properties of solution sets.**QUIZ 1.****Jan. 19th**: §1.3, cont.: Properties of solution sets. Homogeneous systems of equations. §1.5: vectors and vector operations.**Jan. 22nd**: §1.5, cont.: Matrix operations. Matrix multiplication is function composition.**Jan. 24th**: §1.6: Properties of matrix operations. Transposes, miscellaneous notation.**Jan. 26th**:**QUIZ 2.**§1.7: Linear combinations.**Jan. 29th**: §1.7, cont.: Linear independence and nonsingular matrices.**Jan. 31st**: §1.9: Matrix inverses: intuition and proofs.**Feb. 2nd**:**QUIZ 3**. §1.9, cont.: Proof that nonsingular matrices have inverses, and review of nonsingular matrices.**Feb. 5th**: §1.9, cont.: Computing matrix inverses; properties of inverses.**Feb. 7th**: §1.6: Dot product and norm. Chapter 2: Lines and planes in ℝ^{2}and ℝ^{3}.**Feb. 9th**:**MIDTERM I.****Feb. 12th**: §3.1–3.2: Subspaces of ℝ^{n}: examples and non-examples.**Feb. 14th**: §3.3: Subspaces of ℝ^{n}: span of a set of vectors, range of a matrix.**Feb. 16th**:**QUIZ 4**. §3.3: Subspaces of ℝ^{n}: row space and null space (a.k.a. kernel) of a matrix.**Feb. 19th**: More about the null space. §3.4: Bases for subspaces.**Feb. 21st**: §3.4: Computing bases for subspaces; building bases vector by vector.**Feb. 23rd**:**QUIZ 5**. §3.5: Dimension.**Feb. 26th**: §3.5, cont.: Rank and the rank–nullity theorem.**Feb. 28th**: §3.6: Orthogonal and orthonormal bases for subspaces.**March 2nd**:**QUIZ 6**. §3.6, cont.: Orthogonal projection and the Gram–Schmidt process.**March 5th**: §3.6, cont.: More on orthogonal projection. Intro to least squares regression (§3.8.)**March 7th**: §5.1–5.3: Examples of (abstract) vector spaces and their subspaces.**March 9th**:**QUIZ 7**. §5.3–5.4: Span, linear independence, and bases for abstract vector spaces.**March 12th–16th**:**SPRING BREAK!****March 19th**: §3.7: Linear transformations (a coordinate-free way of thinking about matrices). Geometric examples.**March 21st**:**QUIZ 8**. §3.7, cont.: More examples of linear transformations. Interpreting matrices geometrically.**March 23rd**: §3.7, cont., and 4.1: Methods of geometrically interpreting matrices.**March 26th**: §4.1, cont.: Definition and examples of eigenvectors and eigenvalues for 2×2 matrices.**March 28th: MIDTERM II.****March 30th**: §4.1, cont.: Computing eigenvectors and eigenvalues for 2×2 matrices. §4.2: The concept of determinant.**April 2**: §4.2: Computing determinants.**April 4**:**QUIZ 9**. §4.4: Eigenvalues of*n*×*n*matrices (*n*≥ 3) and the characteristic polynomial.**April 6**: §4.4: Some facts about eigenvalues. §4.5: Eigenvectors and their linear independence.**April 9**: §4.5: The meaning of repeated eigenvalues.**April 11**:**QUIZ 10**. §4.6: Review of complex numbers.**April 13**: §4.6: The meaning of complex eigenvalues. Complete classification of 2×2 matrices; similar matrices, segueing into §4.7.**April 16**: §4.7: Similar matrices and diagonalization.**April 18**: §4.7: Diagonalization (and partial diagonalization).**Upcoming**: §4.7: Orthogonal matrices and the spectral theorem.