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- Topic 1. Matrix algebra and vector algebra.
- Appendix 1: Miscellanies on theoretical mathematics.
- Appendix 2: Words and phrases indicating logical content of statements.
- Exercise for 1.3 (Answer)
- Appendix 1: Mathematical induction.
- Appendix 2: Dis-proof by providing a counter-example.
- Exercise for 1.4 (Answer)
- Topic 2. Systems of linear equations.
- Topic 3. Invertible matrices
- Everything in the previous topics will be assumed, and used freely where necessary.
- Topic 4. Spaces, bases, and dimensions.
- Everything in the previous topics will be assumed, and used freely where necessary.
- Topic 5. Determinants.
- Everything in Topics 1-3 will be assumed, and used freely where necessary.
- Topic 6. Eigenvalues, eigenvectors, and diagonalization.
- Everything in Topics 1-5 will be assumed, and used freely where necessary.
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Further material | ||
---|---|---|---|---|---|---|---|
1 | 1 | Matrices, matrix addition, and scalar multiplication for matrices. | Notes | Week 1 Tuesday | Exercise for 1.1 (Answer) | ||
1 | 2 | Matrix multiplication. | Notes | Week 1 Tuesday | 1.1. | Exercise for 1.2 (Answer) | |
1 | 3 | Transpose, symmetry and skew-symmetry. | Notes | Week 1 Thursday | 1.1, 1.2. | ||
1 | 4 | Commuting matrices versus non-commuting matrices. | Notes | Week 2 Tuesday | 1.1, 1.2. | ||
1 | 5 | Linear combinations. | Notes | Week 2 Tuesday | 1.1, 1.2, 1.3. | Exercise for 1.5 (Answer) | |
1 | 6 | Linear dependence and linear independence. | Notes | Week 3 Tuesday | 1.5, 2.1. | Exercise for 1.6 (Answer) | |
1 | 7 | Row operations on matrices. | Notes | Week 3 Thursday | 1.1. | Exercise for 1.7 (Answer) | |
1 | 8 | Row operations and matrix multiplication. | Notes | Week 4 Tuesday | 1.2, 1.7. | Exercise for 1.8 (Answer) |
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Further material | ||
---|---|---|---|---|---|---|---|
2 | 1 | Systems of linear equations. | Notes | Week 2 Thursday | 1.1, 1.2. | Exercise for 2.1 (Answer) | |
2 | 2 | Row-echelon forms and reduced row-echelon forms. | Notes | Week 3 Tuesday | 2.1, 1.5 (preferred), 1.6 (preferred). | ||
2 | 3 | Existence of reduced row-echelon form row-equivalent to given matrix, (and the uniqueness question). | Notes | Week 4 Tuesday | 1.7, 1.8, 2.2. | ||
2 | 4 | Solving systems of linear equations. | Notes | Week 4 Thursday | 2.1, 2.2, 2.3, 1.5 (preferred), 1.6 (preferred). | Exercise for 2.4 (Answer) | |
2 | 5 | Linear combinations from the point of view of systems of linear equations. | Notes | Week 5 Tuesday | 1.5, 2.4. | Exercise for 2.5 (Answer) | |
2 | 6 | Linear dependence and linear independence from the point of view of homogeneous systems of linear equations. | Notes | Week 5 Tuesday | 1.6, 2.4. | Exercise for 2.6 (Answer) |
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Supplementary material | ||
---|---|---|---|---|---|---|---|
3 | 1 | Invertible matrices. | Notes | Week 5 Thursday | Exercise for 3.1 (Answer) | ||
3 | 2 | Invertibility and row operations. | Notes | Week 6 Tuesday | 3.1. | Exercise for 3.2 (Answer) | |
3 | 3 | Various necessary and sufficient conditions for invertibility. | Notes | Week 6 Tuesday | 3.1, 3.2. | Exercise for 3.3 (Answer) | |
3 | 4 | Row-equivalence from the point of view of invertible matrices. | Notes | Week 6 Thursday | 3.2. |
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Supplementary material | ||
---|---|---|---|---|---|---|---|
4 | 1 | Sets of matrices and sets of vectors. | Notes | Week 7 Tuesday | |||
4 | 2 | Set equality (for sets of matrices and sets of vectors). | Notes | Week 7 Tuesday, Week 7 Thursday | 4.1. | ||
4 | 3 | Subspaces of column vectors. | Notes | Week 8 Tuesday | 4.1, 4.2. | ||
4 | 4 | Basis for subspaces of column vectors. | Notes | Week 8 Tuesday | 4.1, 4.2, 4.3. | Appendix: Proof of the `Change-of-basis' Theorem. | |
4 | 5 | Dimension for subspaces of column vectors. | Notes | Week 8 Thursday | 4.1, 4.2, 4.3, 4.4. | Appendix: Proofs of basic theoretical results concerned with bases and dimensions for general subspaces of column vectors. | |
4 | 6 | Span, column space, row space, and minimal spanning set. | Notes | Week 9 Tuesday, Week 9 Thursday. | 4.2, 4.3, 4.4, 4.5. | Appendix: Proofs of basic theoretical results concerned with column space, row space, and minimal spanning set. | |
4 | 7 | Basis and dimension, null space, and the notion of subspace of a subspace. | Notes | Week 9 Thursday, Week 10 Tuesday. | 4.4, 4.5, 4.6. | Appendix: Proofs on theoretical results about dimension for general subspaces of R^{n}. | |
4 | 8 | Rank-nullity Formulae. | Notes | Week 10 Tuesday | 4.5, 4.6, 4.7. | Exercise for 4.1-4.8 (Answer) |
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Supplementary material | ||
---|---|---|---|---|---|---|---|
5 | 1 | Determinants of square matrices. | Notes | Week 11 Tuesday, Week 11 Thursday | Appendix: Expansion of determinants along arbitrary rows. | ||
5 | 2 | Row operations and determinants. | Notes | Week 11 Thursday, Week 12 Tuesday | 5.1 | ||
5 | 3 | Transpose, column operations and determinants. | Notes | Week 12 Tuesday | 5.1, 5.2. |
Topic | Sub-topic | Title | Schedule (for MATH1030B) | Sequel to something? | Supplementary material | ||
---|---|---|---|---|---|---|---|
6 | 1 | Eigenvalues and eigenvectors for square matrices. | Notes | Week 12 Tuesday, Week 12 Thursday | |||
6 | 2 | Diagonalization and diagonalizability for square matrices. | Notes (Revised 29/11.) | Week 13 Tuesday | 6.1 | ||
6 | 3 | Eigenvalues and characteristic polynomials. | Notes (Revised 29/11.) | Week 13 Tuesday | 6.1, 6.2. | Exercise for 6.1-6.3 (Answer) |