Course Information

Lectures: MWF at 9am in Art 204 by Prof. Casals. Attendance not mandatory. Facilities did not provide this lecture room with video capturing technology, so lectures cannot be video recorded. The Math Diaries below will contain precise details on the content of each lecture, which textbook section is reading assigned to each lecture and, when possible, lecture notes.

Recitations: T at 4:10pm (Section 1) and 5:10pm (Section 2) in Bainer Hall 1128 by TA Scroggin. Attendance not mandatory.

Textbook: Linear Algebra As An Introduction To Abstract Mathematics by I. Lankham, B. Nachtergaele and A. Schilling. If you want to start reading before the course starts, please try reading Chapters 1 and 4.

Syllabus: The course syllabus for course guidelines. Please do read the syllabus carefully, many questions you might have are answered in the syllabus.

Office Hours and Contact Information: The office hours of Professor Casals (casals--at--ucdavis.edu) are Mondays 2-3pm and Wednesday 2-3pm at MSB 3214. Always feel free to ask after class. The office hours for Tonie Scroggin will be available soon.

Piazza: The Piazza for this course is "MAT 067 A01-A02 SQ 2024" (https://piazza.com/ucdavis/spring2024/mat067a01a02sq2024/info). You can access the Piazza forum through the UC Davis Canvas MAT-67 course.

Important Dates: First Day (Apr 1), First Midterm Test (Apr 26), Second Midterm Test (May 31), Final Exam (June 13).

Additional dates: Add Date (Apr 16), Drop Date (Apr 26), Last Day of Instruction (June 6).

Problem Sets: Weekly assignments due on Fridays at 9am through Gradescope.

Problem Sets and Solutions

1. Problem Set 1: Due Friday Apr 12, available Apr 4. (TeX file)
   
   
2. Solutions Problem Set 1: available Apr 12.
   
   
3. Problem Set 2: Due Friday Apr 19, available Apr 12. (TeX file)
   
   
4. Solutions Problem Set 2: available Apr 22.
   
   
5. Practice 1st Midterm I: no due date.
   
   
6. Solutions Practice 1st Midterm I: no due date.
   
   
7. Practice 1st Midterm II: no due date.
   
   
8. Solutions Practice 1st Midterm II: no due date.
   
   Practice Problems for Lecture 1
   Solutions Practice Problems for Lecture 1
   
   Practice Problems for Lecture 2
   Solutions Practice Problems for Lecture 2
   
   Practice Problems for Lecture 3
   Solutions Practice Problems for Lecture 3
   
   Practice Problems for Lecture 4
   
   Practice Problems for Lecture 5
   
   Practice Problems for Lecture 6
   
   Practice Problems for Lecture 7
   
   Practice Problems for Lecture 8
   
   

Math Diaries

1. Monday Apr 1: Introduction to MAT-67.
   The goal of Linear algebra: linear systems of equations.
   Examples and non-examples of linear systems of equations.
   Applications: Taylor series, diagonalization, PageRank & graph theory.
   From systems of equations to maps. Definition of a map. Examples and non-examples.
   
   Textbook Reading (Apr 1): Chapter 1.
   Lecture 1 Notes: Notes.
   Practice Problems for Lecture 1.
   Solutions Practice Problems for Lecture 1.
   
   
   
2. Wednesday Apr 3: Linear Maps.
   Review of maps. Definition of linear maps.
   Examples and non-examples of linear maps.
   Proof that composition of linear maps is linear.
   
   Textbook Reading (Apr 3): Section 1.3.
   Lecture 2 Notes: Notes.
   Practice Problems for Lecture 2.
   Solutions Practice Problems for Lecture 2.
   
   
   
3. Friday Apr 5: The geometry of linear Maps.
   How to draw ℝ and ℝ². Classification of linear maps from ℝ to ℝ.
   Drawing a linear maps from ℝ² to ℝ². f(1,0) and f(0,1) suffice to draw everything.
   Rotations in ℝ². Composition of rotations.
   
   Textbook Reading (Apr 5): Chapter 1 (but not covered much).
   Lecture 3 Notes: Notes.
   Practice Problems for Lecture 3.
   Solutions Practice Problems for Lecture 3.
   
   
   
4. Monday Apr 8: Vector spaces.
   Properties of vector sum in ℝⁿ. Properties of scalar multiplication in ℝⁿ
   Definition of a vector space over a field k. Discussion of geometric intuition.
   Examples of vector spaces: ℝⁿ, Qⁿ. Spaces of polynomials.
   Non-examples: Q as an ℝ-v.s., polynomials of fixed degree.
   
   Textbook Reading (Apr 8): Chapter 4.1.
   Lecture 4 Notes: Notes.
   Practice Problems for Lecture 4.
   
   
   
5. Wednesday Apr 10: Vector subspaces.
   Recall: vector spaces and running examples (ℝⁿ and ℝ[x]).
   Sum and scalar multiplication preserve solutions of homogeneous linear systems.
   Definition of a vector subspace. Discussion of geometric intuition.
   Examples and non-examples of vector subspaces: in ℝⁿ and ℝ[x].
   
   Textbook Reading (Apr 10): Chapter 4.3.
   Lecture 5 Notes: Notes.
   Practice Problems for Lecture 5.
   
   
   
6. Friday Apr 12: Sums, intersections and direct sums.
   The sum of two subspaces. Examples.
   The intersection of two subspaces. Examples.
   The issue with non-unique decomposition in sum.
   Definition of direct sums and examples.
   
   Textbook Reading (Apr 12): Chapter 4.4.
   Lecture 6 Notes: Notes.
   Practice Problems for Lecture 6.
   
   
   
7. Monday Apr 15: Spans and basis.
   Linear combinations: definitions and examples.
   Linear span: definition and examples.
   Properties of span subspaces. Definition of dimension.
   Linear independence. Examples.
   
   Textbook Reading (Apr 15): Chapter 5.1 and 5.2.
   Lecture 7 Notes: Notes.
   Practice Problems for Lecture 7.
   
   
   
8. Wednesday Apr 17: Spans and linear dependence.
   Vectors in span are unique linear combinations (Lemma 5.2.6).
   Linear dependence lemma (5.2.7). Examples.
   Spans are the smallest subspaces containing the vectors. (Theorem 5.2.9).
   
   Textbook Reading (Apr 17): Chapter 5.2.
   Lecture 8 Notes: Notes.
   Practice Problems for Lecture 8.
   
   
   
9. Friday Apr 19: Basis.
   
   L.i. vectors spanning are the smallest possible spanning sets (Thm 5.2.9).
   Definition of a basis. Several examples. Basis Extension Theorem (Thm 5.3.7).
   Thm 5.4.2 and definition of dimension. Thm 5.4.4.
   
   Textbook Reading (Apr 19): Chapter 5.3 and 5.4 (up to Thm 5.4.4).
   Lecture 9 Notes: Notes.
   Practice Problems for Lecture 9 on Monday Apr 22.
   
   
   
10. Friday Apr 22: Spans, linear independence & Basis.
    
    Conceptual exercises on spanning sets and linear dependence.
    Example on finding a basis for a subspace given by equation.
    Finding basis for vector subspaces of polynomials.
    
    Textbook Reading (Apr 22): Chapter 5 (up to Thm 5.4.4).
    Lecture 10 Notes: Notes.