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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.
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 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).
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.
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 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.

• 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)

• 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.