Linear Algebra I (MATH1030B)

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The schedule is tentative.

• Topic 1. Systems of linear equations.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
1	1	What is solving a system of linear equations?	Notes		Week 1 Tuesday		Further examples. Further and advanced examples.
1	2	Basic terminologies on systems of linear equations.	Notes		Week 1 Tuesday
1	3	Row operations on matrices.	Notes	Annotated Plates	Week 1 Thursday		Further examples
1	4	Augmented matrix representation.	Notes		Week 2 Tuesday	1.1, 1.2, 1.3.
1	5	Row-echelon forms and reduced row-echelon forms.	Notes	Annotated Plates	Week 2 Tuesday	1.4.	Supplementary illustration 1. Supplementary illustration 2.
1	6	Classification of systems of linear equations in terms of reduced row-echelon forms.	Notes		Week 2 Tuesday	1.5.

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• Topic 2. Matrices and vectors.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
2	1	Addition and scalar multiplication for matrices and vectors.	Notes		Week 2 Thursday		Supplementary illustration.
2	2	Matrix multiplication.	Notes	Annotated Plates	Week 3 Tuesday	2.1.
2	3	Vector presentation and matrix presentation for a system of linear equations and its solutions.	Notes		Week 3 Tuesday	2.2.
2	4	Row operations and matrix multiplication.	Notes	Annotated Plates	Week 3 Tuesday	1.3, 2.2.
2	5	Miscellanies on matrices.	Notes	Annotated Plates	Week 3 Thursday	2.1, 2.2.
2	6	Examples of simple proofs in linear algebra.	Notes	Annotated Plates	Week 4 Tuesday	2.5.
2	7	Further miscellanies on mathematical reasoning in linear algebra.	Notes		Self-study supplementary material	2.5, 2.6.

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• Topic 3. Solution sets

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
3	1	Set notations in linear algebra.	Notes	Annotated Plates	Week 4 Tuesday	1.1, 2.3.	Supplementary illustration.
3	2	Homogeneous systems and null spaces.	Notes	Annotated Plates	Week 4 Tuesday	3.1.
3	3	Homogeneous system associated to a system of linear equations.	Notes	Annotated Plates	Week 4 Thursday	3.1, 3.2.	Supplementary illustration 1. Supplementary illustration 2.
3	4	Geometry of solution sets for systems of linear equations.	Notes		Self-study supplementary material	3.1.	Supplementary illustration 1. Supplementary illustration 2.

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• Topic 4. Nonsingularity.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
4	1	Non-singular matrices.	Notes	Annotated Plates	Week 5 Thursday	3.2.
4	2	Non-singularity in terms of row operations and matrix multiplication.	Notes	Annotated Plates	Week 6 Tuesday	4.1.
4	3	Non-singularity and invertibility.	Notes	Annotated Plates	Week 6 Tuesday	4.2.	Further examples.
4	4	Determining invertibility and computing matrix inverse through row operations.	Notes	Annotated Plates	Week 6 Tuesday	4.3.	Further examples.
4	5	Existence and uniqueness of solutions for a system of linear equations whose coefficient matrix is a square matrix.	Notes	Annotated Plates	Week 6 Thursday	4.4.
4	6	Row equivalence in terms of multiplication by non-singular and invertible matrices.	Notes	Annotated Plates	Week 6 Thursday	4.4.

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• Topic 5. Linear combinations.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
5	1	Linear combinations.	Notes	Annotated Plates	Week 7 Tuesday	2.1, 2.2, 4.3.
5	2	Span of vectors and column space of a matrix.	Notes	Annotated Plates	Week 7 Tuesday	3.1, 5.1.
5	3	Subspaces of Rn.	Notes	Annotated Plates	Week 7 Tuesday	3.2, 5.2.
5	4	Subspaces of Rn in contrast general subsets of Rn.	Notes	Annotated Plates	Week 7 Thursday	5.3.	Supplementary illustration.
5	5	How to determine whether a given vector is the linear combination of some other vectors.	Notes	Annotated Plates	Week 8 Tuesday	1.5, 1.6, 5.1.
5	6	Linear dependence and linear independence.	Notes	Annotated Plates	Week 8 Tuesday	3.2, 5.1.	Further examples.
5	7	Theoretical results on linear dependence and linear independence.	Notes	Annotated Plates	Week 8 Tuesday	4.3, 5.6.	Supplementary illustration. Further examples.
5	8	Theoretical results on span of vectors and column space of a matrix.	Notes	Annotated Plates	Week 8 Thursday	4.3, 5.2.
5	9	How to express the column space of a matrix as the null space of some matrix.	Notes		Self-study supplementary material	1.6, 2.2, 3.2, 4.6, 5.2.

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• Topic 6. Basis and dimension.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
6	1	Bases for subspaces of Rn.	Notes	Annotated Plates	Week 9 Tuesday	5.1, 5.2, 5.3, 5.6, 5.7.	Further examples.
6	2	Gaussian elimination and basis for null space.	Notes	Annotated Plates	Week 9 Tuesday	1.5, 3.2, 6.1.
6	3	Minimal spanning sets.	Notes	Annotated Plates	Week 9 Tuesday	5.1, 5.7, 5.8, 6.1.	Further examples.
6	4	More on minimal spanning sets.	Notes	Annotated Plates	Week 9 Thursday	6.3.
6	4A	Replacement Theorem (as Appendix to `More on minimal spanning sets').	Notes		Self-study supplementary material	5.7, 5.8, 6.1.
6	5	How to obtain a basis for the intersection of two subspaces of Rn.	Notes		Self-study supplementary material	5.9, 6.2.
6	6	Transpose and row space.	Notes	Annotated Plates	Week 9 Tuesday	4.3, 5.8, 6.1.	Further examples.
6	6A	Duality between spanning and linear independence (as Appendix to `Transpose and row space').	Notes		Self-study supplementary material	6.6.
6	7	Dimension.	Notes		Week 10 Tuesday	6.1, 6.4.
6	8	Inequalities on dimension.	Notes	Annotated Plates	Week 10 Tuesday	6.4, 6.7.
6	8A	How to check whether some given vectors constitute a basis for a given subspace of Rn (as Appendix to `Inequalities on dimension').	Notes		Self-study supplementary material	5.6, 6.3, 6.8.
6	9	Rank-nullity Formula.	Notes	Annotated Plates	Week 10 Tuesday	6.2, 6.3, 6.6, 6.7, 6.8.
6	9A	Dimension relation on sum and intersection (as Appendix to `Rank-nullity Formula').	Notes		Self-study supplementary material	6.4, 6.7, 6.8, 6.9.
6	9B	How bases for the same subspace of Rn relate to each other (as Appendix to `Rank-nullity Formula').	Notes		Self-study supplementary material	6.1, 6.7, 6.9.

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• Topic 7. Eigenvalues, eigenvectors and determinants.
• Everything in the previous topics will be assumed, and used freely where necessary.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
7	1	Eigenvalue and eigenvector.	Notes	Annotated Plates	Week 10 Thursday
7	2	Diagonalizability.	Notes	Annotated Plates	Week 11 Tuesday	7.1.
7	3	Determinants.	Notes	Annotated Plates	Week 11 Thursday
7	4	Multilinearity and Alternating property.	Notes	Annotated Plates	Week 12 Tuesday	7.3.	Further examples. Further advanced examples.
7	5	Products of determinants.	Notes	Annotated Plates	Week 12 Tuesday	7.4.	Further examples.
7	6	Characteristic polynomial of a matrix.	Notes	Annotated Plates	Week 12 Tuesday	7.5.	Further examples.
7	7	Adjoint of a matrix.	Notes		Self-study supplementary material	7.5.	Illustrations

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• Topic 8. Inner-product.
• Everything in the previous topics will be assumed, and used freely where necessary.

Topic	Handout	Title			Schedule	Sequel to something?	Supplementary material
8	1	Inner product, norm, and orthogonality.	Notes	Annotated Plates	Week 13 Tuesday
8	2	Orthogonal complement.	Notes		Self-study supplementary material	8.1.
8	3	Orthonormal bases and orthogonal projections.	Notes	Annotated Plates	Week 13 Tuesday	8.1, 8.2.	Further examples.
8	4	Gram-Schmidt orthogonalization process.	Notes	Annotated Plates	Week 13 Thursday	8.3.	Further examples.
8	5	Orthogonal matrices.	Notes		Self-study supplementary material	8.4.

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