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| Date | Topics |
| Jan.13. | Matrix encodings of linear systems. Elementary row operations on matrices. |
| Jan.15. | Echelon and reduced echelon forms.
Examples of row reduction to echelon
forms. HW1 (due Wed. Jan. 22.): 1.1: 4, 10, 12, 20, 24; 1.2: 2, 8, 10, 16, 18, 22, 26. |
| Jan.17. | The row reduction algorithm. Pivots. Existence and uniqueness of solutions to linear systems. |
| Jan.20. | Martin Luther King Holiday; no class. |
| Jan.22. | Linear combinations of vectors. Spans.
HW1 (due Fri. Jan. 31.): 1.2: 20, 24, 28; 1.3: 8, 10, 12, 18, 22, 33; 1.4: 10, 15, 18, 26. |
| Jan.24. | Matrix-vector product. Vector equations and matrix equaitons. |
| Jan.27. | Two existence theorems on solutions of matrix equations. |
| Jan.29. | Homogeneous matrix equations. Solutions
of homogeneous vs non-homogeneous
equations. HW3 (due Wed. Feb.5): 1.4: 25, 27, 36; 1.5: 4, 16, 24, 26, 28, 36; 1.7: 6, 12, 20, 32. |
| Jan.31. | Linear independence. |
| Feb.03. | More on linear independence. |
| Feb.05. | Definition of linear transformations.
Connections to matrix
multiplicaitons. HW4 (due Wed. Feb.12): 1.7: 36, 37, 38; 1.8: 6, 15, 16, 24, 32, 36; 1.9: 6, 8, 14, 22. |
| Feb.07. | Consequences of linearity. The matrix of a linear map. |
| Feb.10 | Geometric linear transformations in R^2. |
| Feb.12. | Image and surjectivity; kernel
and injectivity.
HW 5 (due Wed. Feb.19): 1.9: 2, 14, 26, 28, 36; 2.1: 2, 9, 10, 16, 20, 22, 24. |
| Feb.14. | Scaling, addition and transposition of matrices. Properties of these operations. |
| Feb.17. | Matrix multiplication and its properties. |
| Feb.19. | More properties of matrix multiplication.
HW6 (due Wed. Feb.26): 2.1: 27, 28, 30; 2.2: 4, 6, 14, 16, 18, 22, 30, 32. |
| Feb.21. | Invertible matrices: definition, properties, and relation to linear maps. Review for Midterm 1 |
| Feb.24. | Characterization of invertible matrices. Finding matrix inverses. |
| Feb.26. | Review for Midterm 1. HW7 (due Wed. Mar.4): 2.3: 6, 8, 13, 16, 18, 34. |
| Feb.28. | Midterm 1 (Solutions) |
| Mar.02. | Subspaces of vector spaces: definition and examples. |
| Mar.04. | Subspaces arising from matrices and
maps. Bases. HW8 (due Wed. Mar.11.): 2.8: 1, 4, 7, 16, 18, 22, 24, 26, 28, 30. |
| Mar.06. | Bases of R^n. Bases of null spaces and column spaces. |
| Mar.09. | Bases verification and computation. |
| Mar.11. | More bases computation. The rank-nullity
theorem. HW9 (due Fri. Mar.20.): 2.9: 10, 14, 15, 16, 20, 22, 23, 26; 3.1: 2, 4, 12, 14. |
| Mar.13. | Class format after transition to remote teaching. Determinants of matrices of small size. |
| Mar.16. | Cofactor expansions. |
| Mar.18. | Determimants, row operations and
invertibility. HW10 (due Mon. Mar.30.): 3.2: 18, 20, 22, 24, 28, 29, 38, 40. 3.3: 20, 22, 28. |
| Mar.20. | Determinants of matrix products and transposes. Geometry of determinants. |
| Mar.30. | Definition of abtract vector spaces. Spans, linear independence, subspace, and linear transformations, revisited. |
| Apr.01. | Images and kernels. Bases and dimensions.
HW11 (due Fri.Apr.10.): 4.1: 6, 8, 10, 12, 20, 22, 28; 4.2: 10, 12, 18; 4.3. 14, 20; 4.5: 4, 8, 14. |
| Apr.03. | Coordinate vectors. Coordinate mappings. Properties of linear isomorphisms. |
| Apr.06. | Row space. Rank. Homework guide. |
| Apr.08. | Change of basis. HW12 (due Wed. Apr.15.): 4.4: 4, 6, 14, 17, 23, 28, 34; 4.7: 2, 4, 6, 8, 14, 15. |
| Apr.10. | Finding change-of-basis matrices. Review for Midterm 2. |
| Apr.13. | Matrix encodings of linear maps. |
| Apr.15. | Midterm 2 (Solutions)
HW13 (due Fri. Apr.24.): 5.1: 2, 6, 10, 14, 18, 24; 5.2: 6, 12, 16; 5.4: 2, 4, 6. |
| Apr.17. | Eigenvectors and eigenvalues. |
| Apr.20. | Eigenspaces as null spaces. Finding eigenvalues via characteristic equations. |
| Apr.22. | Complex eigenvalues. Algebraic vs.
geometric multiplicities. Eigenbases.
HW14. (due Fri. May.1.): 5.2: 18; 5.3: 2, 6, 10, 12, 18, 24; 5.4: 12, 16, 17, 18. |
| Apr.24. | Diagonalizability and diagonalization. |
| Apr.27. | Inner products, lengths and distances. |
| Apr.29. | Orthogonality. Orthogonal sets. Orthogonal
projections.
HW15. (to be completed, but will not be collected): 6.1: 2, 6, 10, 14, 22, 24, 28; 6.2: 6, 7, 9, 12, 14. |
| May.06. | Final Exam, 1:30 - 4:15 p.m. on Canvas. |