Linear Algebra I (MATH1030B)

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• Topic 1. Matrix algebra and vector algebra.

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.	Appendix 1: Miscellanies on theoretical mathematics. Appendix 2: Words and phrases indicating logical content of statements. Exercise for 1.3 (Answer)
1	4	Commuting matrices versus non-commuting matrices.	Notes		Week 2 Tuesday	1.1, 1.2.	Appendix 1: Mathematical induction. Appendix 2: Dis-proof by providing a counter-example. Exercise for 1.4 (Answer)
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)

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• Topic 2. Systems of linear equations.

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).	Appendix: Reduced row-echelon forms of small sizes. Exercise for 2.2 (Answer)
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.	Appendix: Proof of existence of row-echelon form and reduced row-echelon form row-equivalent to given matrix. Exercise for 2.3 (Answer)
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)

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• Topic 3. Invertible matrices
• Everything in the previous topics will be assumed, and used freely where necessary.

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.	Appendix: Uniqueness of reduced row-echelon form row-equivalent to given matrix. Exercise for 3.4 (Answer)

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• Topic 4. Spaces, bases, and dimensions.
• Everything in the previous topics will be assumed, and used freely where necessary.

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 Rn.
4	8	Rank-nullity Formulae.	Notes		Week 10 Tuesday	4.5, 4.6, 4.7.	Exercise for 4.1-4.8 (Answer)

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• Topic 5. Determinants.
• Everything in Topics 1-3 will be assumed, and used freely where necessary.

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.	Appendix: Co-factors, adjoints, and Cramer's Rule. Exercise for 5.1-5.3 (Answer)

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• 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?	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)

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