Please see below for lecture summaries, homework and other study material. For your grades, please see Canvas.
| Date | Topics |
| Aug.23. | Matrix encodings of linear systems. Elementary row operations on matrices. Notes. |
| Aug.25. | Echelon and reduced echelon forms.
Examples of row reduction.
Notes.
HW1 (due Wed. Sep.01.): 1.1: 4, 12, 16, 20, 22, 24; 1.2: 2, 8, 10, 16, 18. |
| Aug.27. | The Gaussian elimination algorithm. Pivots. Notes. |
| Aug.30. | Existence and uniqueness of solutions for linear systems via Echelon forms. Notes. |
| Sep.01. | Matrix-vector products. Linear systems
as vector equations and matrix equations. Notes.
HW2 (due Wed. Sep.08.): 1.3: 2, 6, 10, 14, 16, 21, 22, 24, 26. |
| Sep.03. | Spans and spanning sets. Notes. |
| Sep.06. | Labor day; no class. |
| Sep.08. | Proof of echelon form criterion for
spanning sets. Linear independence.
Notes.
HW3 (due Wed. Sep.15.): 1.4: 8, 10, 13, 14, 15, 22, 25, 26; 1.5: 16, 18. |
| Sep.10. | More criteria for linear independence, with proofs. Geometry of R^2 and R^3. Notes. |
| Sep.13. | The parallellogram law. Solution sets of homogeneous vs. nonhomogeneous equations. Notes. |
| Sep.15. | Linear transformations: definition, properties, and examples.
Notes.
HW4 (due Wed. Sep.22.): 1.5: 25, 26; 1.7: 4, 14, 16, 18, 20; 1.8: 10, 17, 20. |
| Sep.17. | Relationship between linear maps and matrix multiplications. Image, kernel, surjectivity and injectivity of linear maps.
Notes.
Midterm I date announced: Friday, October 1 (in class). |
| Sep.20. | Image, surjectivity, and spans; kernel, injectivity and linear independence. Some geometric transformations in R^2. Notes. |
| Sep.22. | More geometric transformations. Addition, scaling and multiplication of matrices.
Notes.
HW5 (due Wed. Sep.29.): 1.9: 4, 6, 8, 10, 12; 2.1: 2, 9, 10. |
| Sep.24. | Properties and "non-properties" of matrix operations. Notes. Review for Midterm I. |
| Sep.27. | Matrix multiplication vs. composition of linear maps. Proof of associativity for matrix multplication. Definition of invertible matrix. Notes. |
| Sep.29. | Inversion and cancellation. Inverses of 2*2 matrices. Powers and transposes of matrices. Midterm questions. Notes. (No homework is due next Wednesday.) |
| Oct.01. | Midterm I. |
| Oct.04. | More properties of inversion. Algorithm for tesing invertibility and finding inverses. Notes. |
| Oct.06. | Example computations of matrix inverses. The invertibility theorem.
Notes.
HW6 (due Wed. Oct.13.): 2.1: 11, 18; 2.2: 4, 7, 25, 32; 2.3: 4, 6, 17, 18, 33, 34. |
| Oct.08. | Invertible linear maps. Subspaces of R^n. Notes. |
| Oct.11. | More examples of subspaces of R^n (with proofs). Notes. |
| Oct.13. | Bases and dimension: definition and first examples. Notes.
HW7 (due Wed. Oct.20.): 2.8: 2, 6, 12, 16, 18; 2.9: 2, 4, 8, 10, 14. |
| Oct.15. | Basis and unique decomposition. Bases and dimensions of the column, row and null spaces of matrices. Notes. |
| Oct.18. | Decomposition vectors. More bases computations. The rank-nullity theorem. Notes. |
| Oct.20. | Computation of determinants via cofactor expansions. Notes.
HW8 (due Wed. Oct.27.): 3.1: 4, 10, 14, 20, 24, 34, 36; 3.2: 16, 18, 20, 22, 24. |
| Oct.22. | Properties of determinants. Notes. |
| Oct.25. | New determinants from old. Applications of determinants. Notes. |
| Oct.27. | Determinants as area/volume.
Definition and examples of abstract vector spaces.
Notes.
HW9 (due Wed. Nov.3.): 3.3: 20, 22; 4.1: 2, 6, 8, 12, 16, 18, 26, 28. |
| Oct.29. | Working with vector space axioms. Linear combinations. Subspaces. Notes.
Midterm II date announced: Friday, November 12 (in class). |
| Nov.01. | Old to new: generalizing familiar notions/facts from R^n to abstract vector spaces. Notes. Proof worksheet. |
| Nov.03. | New to old: solving problems about abstract vector spaces using R^n via
coordinate mappings. Notes. Review for Midterm II.
HW10 (due Wed. Nov.10.): 4.2: 2, 4, 8, 10, 14; 4.4: 2, 8, 12, 14, 28, 32; 4.5: 12, 14. |
| Nov.05. | Going over Homework 10, the proof worksheet, and the midterm review. Notes. |
| Nov.08. | Change-of-basis matrices: definition and properties. Notes. Solutions to review for Midterm II. |
| Nov.10. | Computation of change-of-basis matrices. Matrices of linear maps. Notes. (No homework is due next Wednesday.) |
| Nov.12. | Midterm II. |
| Nov.15. | Motivation for diagonalization. Definition of eigenvalues and eigenvectors. Characteristic polynomials. Notes. |
| Nov.17. | Computation of eigenvalues and eigenvetors. Notes.
HW11 (due Wed. Dec.1.): 5.1: 2, 6, 10, 14; 5.2: 4, 10; 5.3: 2, 10, 12, 14. |
| Nov.19. | The diagonalizability theorem. Notes. |
| Nov.22. | Winter Break Week; no class. |
| Nov.29. | Diagonalization examples. Explanation for the equality "A=PDP^{-1}". Notes. |
| Dec.01. | More on "A=PDP^{-1}". Complex eigenvalues. Definition of inner products. Notes.
HW12 (due Wed. Dec.8.): 5.4: 2, 6, 14, 16; 6.1: 2, 6, 8, 10, 13, 14. |
| Dec.03. | Properties of inner products. Geometric notions arising from inner products. Notes. |
| Dec.06. | Orthogonal sets and bases. Notes. |
| Dec.08. | Orthogonal projections. The Gram--Schmidt process. Notes. Review for Chapters 5 & 6.
HW13 (do not hand in): 6.2: 2, 4, 8, 10, 12, 14; 6.3: 2, 6, 8, 10, 12, 16. |
| Dec.14. | Final Exam, from 4:30 to 7:00 pm. |