Please see below for lecture summaries, homework and other study material.
| Date | Topics |
| Aug.26. | Matrix encodings of linear systems. Elementary row operations on matrices. |
| Aug.28. | Echelon form of matrices. Principles of
row reduction to echelon form.
Homework 1 (due Fri. Sept.6.): 1.1: 4, 12, 18, 20, 24; 1.2: 2, 4, 10, 12, 14, 16. |
| Aug.30. | From echelon forms to solution sets of linear systems. The row reduction algorithm. |
| Sep.02. | Labor day; no class. |
| Sep.04. | Practice problems on solving linear
systems. Linear combinations of
vectors. Homework 2 (due Wed. Sep.11.): 1.3: 2, 6, 12, 18; 1.4: 2, 4, 6, 8, 10, 14, 18, 22. |
| Sep.06. | Vector equations and matrix equations. Spans of vectors. |
| Sep.09. | Geometry of vectors. Homogeneous vs. nonhomogeneous equations. |
| Sep.11. | Linear indepedence of vectors. Homework 3 (due Wed. Sep.18.): 1.5: 4, 6, 16, 22, 32; 1.7: 2, 4, 6, 8, 12, 20, 22, 32. |
| Sep.13. | More on linear independence. Definition of linear maps. |
| Sep.16. | The linearity property. Linear maps vs matrix multiplication. |
| Sep.18. | Geometric linear maps on the plane. Homework 4 (due Wed. Sep.25.): 1.8: 8, 12, 14, 16, 20; 1.9: 4, 6, 8, 14, 22, 38, 40. |
| Sep.20. | Image, kernel, injectivity, and
surjectivity of linear maps. Review Problems for Chapter 1 |
| Sep.23. | Matrix operations: addition, scalar multiplication, and matrix multiplication. |
| Sep.25. | Properties of matrix multiplication. Matrix
transpose. Homework 5 (do not hand in): 2.1: 2, 6, 7, 15, 16, 18, 19. |
| Sep.27. | Definition and basic properties of matrix inverses. |
| Sep.30. | Midterm I. |
| Oct.02. | Characterization of invertible matrices. Algorithm for
finding matrix inverses. Homework 6 (due Wed. Oct.9.): 2.2: 4, 8, 32; 2.3: 6, 8; 2.8: 1, 2, 4, 7. |
| Oct.04. | Subspaces of R^n. |
| Oct.07. | Column spaces and null spaces of matrices. Images and kernels of linear maps. |
| Oct.09. | Bases of subspaces of R^n. Homework 7 (due Wed. Oct.16.): 2.8: 18, 19, 20; 2.9: 2, 4, 8, 10, 14. |
| Oct.11. | Dimensions of vector spaces. The rank-nullity theorem. |
| Oct.14. | Coordinate systems. Determinants and cofactor expansions. |
| Oct.16. | Properties of determinants. Homework 8 (due Wed. Oct.23.): 3.1: 2, 10, 12, 38; 3.2: 16, 18, 20, 22; 3.3: 20, 22. |
| Oct.18. | Cramer's rule. Determinants and volume. |
| Oct.21. | Definition and examples of abstract vector spaces. Spans and linear independence. |
| Oct.23. | Subspaces. Linear maps and their kernels
and images. Composing proofs. Homework 9 (due Wed. Oct. 30.): 4.1: 6, 8, 12, 16, 18, 20; 4.2: 10, 12; 4.3: 4, 14, 20. |
| Oct.25. | Bases and dimensions of abstract vector spaces. |
| Oct.28. | Coordinate systems and coordinate mappings. |
| Oct.30. | Row spaces of matrices. Equality of row
rank and column rank. Homework 10 (due Wed. Nov. 6.): 4.4: 10, 12, 14; 4.5: 8, 10; 4.6: 4, 6, 8, 10; 4.7: 2, 4, 8. Review Problems for Chapters 2-4 (solution guide) |
| Nov.01. | Change of basis. |
| Nov.04. | The matrix of a linear map (relative to chosen bases). |
| Nov.06. | Eigenvectors, eigenvalues. Homework 11 (do not hand in): 5.1: 2, 3, 5, 6, 9, 10, 18; 5.2: 1, 2. |
| Nov.08. | Eigenspaces as kernels. Finding eigenspaces. |
| Nov.11. | Midterm II. |
| Nov.13. | Characteristic polynomials. Finding
eigenvalues. Homework 12 (due Wed. Nov. 20.): 5.1: 1, 8, 11, 12, 14; 5.2: 4, 6, 10, 16; 5.3: 2, 4, 7, 8. |
| Nov.15. | Algebraic multiplicity vs. geometric multiplicitiy of eigenvalues. Diagonalizability. |
| Nov.18. | Diagonalization problems. |
| Nov.20. | More diagonalization problems.
Matrices of linear maps and diagonalization.
Homework 13 (due Wed. Dec. 4.): 5.3: 10, 12, 14; 5.4: 2, 10, 14, 16, 18. |
| Nov.22. | Diagonalization problems. |
| Dec.02. | Inner products. Vector length via inner products. |
| Dec.04. | Distance and orthogonality via inner
products. Orthogonal sets. Homework 14 (due Wed. Dec. 11.): 6.1: 4, 10, 12, 14; 6.2: 2, 8, 10, 12, 14; 6.3: 2, 8, 12, 14. |
| Dec.06. | Orthogonal projections. Review Problems for Chapters 5-6 |
| Dec.09. | The Gram-Schmidt process. |
| Dec.11. | Review for final exam. |