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. Problem Set 3: Due Friday May 10, available Apr 30. (TeX file)
   
   
6. Solutions Problem Set 3.
   
   
7. Problem Set 4: Due Friday May 17, available May 10. (TeX file)
   
   
8. Midterm I. In Lecture Apr 26.
   
   
9. Midterm II. In Lecture May 31.
   
   
10. Practice 1st Midterm I: no due date.
    
    
11. Solutions Practice 1st Midterm I: no due date.
    
    
12. Practice 1st Midterm II: no due date.
    
    
13. Solutions Practice 1st Midterm II: no due date.
    
    
14. Practice 2nd Midterm I: no due date.
    
    
15. Solutions Practice 2nd Midterm I.
    
    
16. Practice 2nd Midterm II: no due date.
    
    
17. Solutions Practice 2nd Midterm II.
    
    
18. Practice Problems for Midterm II: no due date.
    
    
19. Solutions Practice 2nd Midterm II.
    
    
20. Solutions Practice 2nd Midterm II.
    
    
21. Practice Final I.
    
    
22. Practice Final II.
    
    

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. Monday 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.
    
    
    
11. Wednesday Apr 24: Review for midterm.
    
    
12. Friday Apr 26: Midterm Exam.
    
    

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13. Monday Apr 29: Linear maps & Bases.
    
    Linear maps. Composition and properties. Examples.
    Some applications. Two main questions on linear maps.
    Theorem 6.1.3: image of basis determines a linear map.
    
    Textbook Reading (Apr 29): Chapter 6.1.
    Lecture 13 Notes: Notes.
    Practice Problems for Lecture 13.
    
    
    
14. Wednesday May 1: From linear maps to matrices.
    
    A linear map and choices of bases give a matrix: construction.
    Example of matrix for linear map from ℝ³ to ℝ² with given bases.
    Example of two matrices for linear map from ℝ³ to ℝ³ with 2 choices of bases.
    
    Textbook Reading (May 1): Chapter 6.1 & 6.6.
    Lecture 14 Notes: Notes.
    
    
    
15. Friday May 3: Kernels and images.
    
    Definitions of kernel and image. Lemma 1: Kernel is a subspace.
    Lemma 2: image is a subspace. Proofs of lemmas. Example.
    Practice problem. Relation to linear systems.
    
    Textbook Reading (May 3): Chapters 6.2 & 6.3.
    Lecture 15 Notes: Notes.
    
    
    
16. Monday May 6: Dimension formula.
    
    The dimension formula. Proof of dimension formula.
    Example of use. Relation to injectivity and surjectivity.
    injectivity and surjectivity in terms of kernels and images.
    
    Textbook Reading (May 6): Chapters 6.4 & 6.5.
    Lecture 16 Notes: Notes.
    
    
    
17. Wednesday May 8: The determinant.
    
    Geometric definition of a determinant. Examples.
    Properties of the determinant. Formula in 2-dimensions and proof.
    More examples. Formula in 3-dimensions.
    
    Textbook Reading (May 8): Chapter 8.2.
    Lecture 17 Notes: Notes.
    
    
    
18. Friday May 10: The determinant II.
    
    Algorithm for computation of determinant in ℝⁿ. Formula and examples.
    Detecting linear dependence via determinants: square case and general case.
    
    Textbook Reading (May 10): Chapter 8.2.
    Lecture 18 Notes: Notes.
    
    
    
19. Monday May 13: Eigenvalues and eigenvectors.
    
    Motivation and definition of eigenvalues. Formula for computing eigenvalues.
    Eigenvectors of given eigenvalue. Examples and computations.
    
    Textbook Reading (May 13): Chapter 7.1 and 7.2.
    Lecture 19 Notes: Notes.
    
    
    
20. Wednesday May 15: Basis of eigenvectors.
    
    Review. When do we have basis made of eigenvectors?
    Theorem 7.2.3: if eigenvalues all different, then we do.
    Proof of Theorem 7.2.3. Examples and non-examples.
    Towards diagonalizations: the matrix of eigenvectors.
    
    Textbook Reading (May 15): Chapter 7.2, 7.3, 7.4 and 7.5.
    
    
    
21. Friday May 17: Diagonalization and invertibility.
    
    Decomposition in termss of diagonal matrices: eigenvector matrix and eigenvalues.
    Computation of the inverse of a matrix in the 2x2 case. Examples.
    Inverse of a matrix in the general case via transpose adjugate.
    
    Textbook Reading (May 17): Chapter 6.7, 7.6 and Section 11.4 (Prop.11.4.2).
    Lecture 21 Notes: Notes.
    
    
    
22. Monday May 20: Applications of diagonalization.
    
    Computation of determinant as product of eigenvalues.
    Computation of powers and exponetial of a matrix.
    Applications to systems of linear differential equations.
    Comments on quantum mechanics and eigenvalues of Hamiltonians.
    
    Textbook Reading (May 20): Section 11.4.
    Lecture 22 Notes: Notes.
    
    
    
23. Wednesday May 22: Inner products.
    
    Motivation and geometric picture. Definition of inner products.
    Examples. Inner products as positive definite symmetric matrices.
    Lemma characterizing inner products in terms of eigenvalues.
    Practice problem and examples on length and orthogonality.
    
    Textbook Reading (May 22): Section 9.1.
    Lecture 23 Notes: Notes.
    
    
    
24. Friday May 24: Inner products and norms.
    
    All inner products are represented by positive definite symmetric matrices.
    Definition of norm. Compatibility of norm with sums and scalar multiplication.
    Pythagorean Theorem (9.3.2). Triangle Inequality (9.3.4). Examples.
    Inner product in the space of functions (see Example 9.1.5).
    
    Textbook Reading (May 24): Sections 9.2 and 9.3.
    Lecture 24 Notes: Notes.
    
    
    
25. Wednesday May 29: Review Midterm II.
    Lecture 25 Notes: Notes.
    
    
    
26. Friday May 31: Midterm Exam II.
    
    

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27. Monday June 3: Fibonacci numbers and golden ratio.
    
    A question on rabbit population. Linear recurrences.
    From a linear recurrence to a linear map (and a matrix).
    Eigenvalues and eigenvectors for Fibonacci recurrence.
    Closed formula for Fibonacci numbers (Binet's formula) using eigenvectors.
    Convergence of consecutive ratio of Fibonacci numbers to golden ratio.
    
    Lecture 26 Notes: Notes.
    
    
    
28. Wednesday June 5: Final review.
    
    Lecture 27 Notes: Notes.