COURSE OUTLINE: MH1201

Course Title

Linear Algebra II

Course Code

MH1201

Offered Study Year 1, Semester 2
Course Coordinator Tan Geok Choo (Dr) gctan@ntu.edu.sg 6513 7452
Pre-requisites MH1200
Mutually exclusive MH2800, MH2802, CY1602
AU 4
Contact hours Lectures: 39, Tutorials: 12
Approved for delivery from AY 2019/20 semester 2
Last revised 8 Jan 2020, 10:17

Course Aims

The course aims to present a careful treatment of the principal topics of linear algebra, such as general vector spaces, linear transformations, diagonalization, and inner product spaces, and to illustrate the power of the subject through a variety of applications. After learning this course, you will be able to make connections between the abstract settings and the concrete problems studied in Linear Algebra I. It is the second of two courses on linear algebra which is a core module for students of mathematical sciences, and a prescribed elective module for students from other schools as well.

Intended Learning Outcomes

Upon successfully completing this course, you should be able to:

  1. Verify a given set V with operations (addition and scalar multiplication) satisfy all axioms of a real or complex vector space, or provide justification if it is not a vector space. Similarly for subspaces, linear transformations, inner product space.
  2. Relate and link the abstract problems to concrete ones, using what have already been studied in Linear Algebra I, and in Calculus I & II.
  3. Verify linear independence and linear combination, a set is a basis, a map is linear, etc.
  4. Prove rigorously the mathematical statements in Linear Algebra: such as properties of vectors spaces, linear transformations, linear independence and basis, eigenvectors and eigenvalues, inner product space
  5. Apply methods of Linear Algebra to different subjects, creating and formulating mathematical models to solve problems in science and engineering involving linear structures.
  6. Apply main theorems and result discussed such as rank nullity theorem.
  7. Compute and interpret eigenvectors and eigenvalues, and eigenspace.
  8. Determine whether a given matrix is diagonalizable and proceed to diagonlize it.
  9. Perform Gram-Schmidt Orthogonalization process to obtain orthogonal basis.

Course Content

General Vector Spaces: Real/Complex Vector Space: Definition and examples; Subspaces.

Basis and Dimension: Linear Combination & Span, Linear Independence, Basis & Coordinate Vectors; Finite Dimensional Space and its Dimension, Basis Construction.

Linear Transformations: Linear Transformation & Examples; Null space, range and Rank-Nullity Theorem; Injection, Surjectiion, Bijection and Isomorhisms.

Matrix Representation of Linear Transformations between finite dimensional vectors spaces: Examples of linear transformations and their matrix representations, Change of basis & Applications.

Eigenvectors & Eigenvalues: Eigenvalues and eigenvectors of Matrices and Linear Transformations, Characteristic Polynomials; Eigenspace, Diagonalization & Applications.

Inner Product Spaces: Inner product and Norm, Inner product space, Some inequalities: Cauchy Schwarz Inequality; Orthogonality, Orthonormal sets, Gram-Schmidt process; Orthogonality, orthogonal projection, orthogonal complement. Applications.

Assessment

Component Course ILOs tested SPMS-MAS Graduate Attributes tested Weighting Team / Individual Assessment Rubrics
Continuous Assessment
Lectures
Test 1 1, 2, 3, 4, 5 1. a, b, c
2. a, b
3. a
15 individual See Appendix for rubric
Test 2 1, 2, 3, 4, 5, 6, 7 1. a, b, c
2. a, b
3. a
25 individual See Appendix for rubric
Examination (2 hours)
Final Examination 1, 2, 3, 4, 5, 6, 7, 8, 9 1. a, b, c
2. a, b
3. a
60 individual See Appendix for rubric
Total 100%

These are the relevant SPMS-MAS Graduate Attributes.

1. Competence

a. Independently process and interpret mathematical theories and methodologies, and apply them to solve problems

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

2. Creativity

a. Critically assess the applicability of mathematical tools in the workplace

b. Build on the connection between subfields of mathematics to tackle new problems

3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

Formative Feedback

Attending lectures, reading lecture slides and related chapters in references, and working through weekly tutorial problems and discussing with instructors and peers, you will be able meet learning outcomes 1,2,3,4,5,6,7,8,9.

After each test, instructors will discuss the test solution to highlight strength and weakness of students.

The Examiner's report will be uploaded to NTULearn to provide feedack on the strength and weakness of students in the module.

Learning and Teaching Approach

Lectures
(39 hours)

Traditional teaching approach: lecture with slides. Theory and examples will be discussed.
Some activities will be conducted during lectures, such as proving and deriving result, verifying axioms or definition, computing eigenvalues or eigenvectors, discussing True or false problems, etc.These will enhance learning of the subjects.

Attend lectures and review lecture notes before coming to class and make sure you understand the last lecture. Missing one lecture may make the next lesson seems like it's in a foreign language.

Regularly summarize the theorems and results, and include examples or methods, in your own words.

Study consistently. Have a study plan (includes the test-dates) and avoid last-minute work or study.

Tutorials
(12 hours)

Linear Algebra II is a continuation of Linear Algebra I, but in a more theoretical approach. It may be hard at the beginning due to the abstract approach. Get with it from the start to avoid having great difficulty later.

Weekly set of problems will be uploaded in NTULearn at least a week in advance before tutorial session so that you have time to read, think and work through the problems to enhance mastery of the topics (both theory and computation).

Tutors will discuss selected problems in each tutorial set. You are encouraged to ask questions to clarify doubts.
You are encouraged to consult with instructors and classmates, and talk about mathematics with your classmates or friends.

Read, think and do the tutorial questions as many as you could. This allows you to understand the abstract concept, the idea and method of proving and computation.

Regularly summarize the theorems and results, and include examples or methods, in your own words. It gets easier as you proceed, with a good understanding of the ideas and concept.

Reading and References

1. H. Anton & C Rorres, Elementary Linear Algebra with Supplemental Applications, Wiley, Eleventh Edition, 2011, ISBN 978-1-118-67745-2

2. J, Fraleigh & Raymond A. Beauregard, Linear Algebra, 3rd Edition, Pearson New International Edition. (Chapters 3,5,6,7,8), ISBN 978-1-29204-272-5.

Course Policies and Student Responsibilities

1. Linear Algebra II is a continuation of Linear Algebra I, but in a more theoretical approach. It may be hard at the beginning due to the abstract approach. Get with it from the start to avoid having great difficulty later.

2. The more you work in the beginning, the less you have to worry later on.

3. Ask questions and consult with your classmates or tutors. Talk and think about mathematics with your classmates or friends.

4. Read, think and do the tutorial questions as many as you could. This allows you have a better understanding of the abstract concept, the idea and method of proving and computation.

5. Attend tutorial sessions to have discussion with tutors and peers.

6. Attend lectures and review your notes before coming to class and make sure you understand the last lecture. Missing one lecture may make the next lesson seems like it's in a foreign language.

7. Regularly summarize the theorems and results, and include examples or methods, in your own words.

8. Study consistently. Have a study plan (includes the test-dates) and avoid last-minute work or study.

9. It gets easier as you proceed, with a good understanding of the ideas and concept.

Academic Integrity

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

As a student, it is important that you recognize your responsibilities in understanding and applying the principles of academic integrity in all the work you do at NTU. Not knowing what is involved in maintaining academic integrity does not excuse academic dishonesty. You need to actively equip yourself with strategies to avoid all forms of academic dishonesty, including plagiarism, academic fraud, collusion and cheating. If you are uncertain of the definitions of any of these terms, you should go to the Academic Integrity website for more information. Consult your instructor(s) if you need any clarification about the requirements of academic integrity in the course.

Course Instructors

Instructor Office Location Phone Email
Tan Geok Choo (Dr) SPMS-MAS-04-12 6513 7452 gctan@ntu.edu.sg

Planned Weekly Schedule

Week Topic Course ILO Readings/ Activities
1

Introduction: Information and Assessment

Chapter 1 General Vector Spaces:
-- Addition, Scalar Multiplication,
- Real and Complex Vectors spaces
- Definition
- Examples
- Results

1, 2, 3, 4, 5, 8

Chapter: Vector Space
Tutorial 1

2

Chapter 2 Subspaces:
- Definition
- Verifying Subspaces
- Properties

Chapter 3 Basis:
Linear combination and span; Related properties and results.

1, 2, 3, 4, 5

Chapters: Subspaces; Basis
Tutorials 1 & 2

3

Linear Independence, Related properties and results.

Basis and
Coordinate Vectors.

Chapter 4 Finite dimensional spaces: Finite dimensional spaces and Dimension.

1, 2, 3, 4, 5

Chapters: Basis; Finite Dimensional Vector Spaces.

Tutorial 2

4

Chapter 4 Finite Dimensional spaces:
- Obtaining basis from Spanning set;
- Basis-extension.
-Dimension of Subspaces

Chapter 5 Linear Transformations:
- Definition & Examples.
- Results

1, 2, 3, 4, 5, 6

Chapters: Basis; Finite Dimensional Vector Spaces.

Tutorial 3

5

Chapter 5 Linear Transformations:
- More Examples.
- Left Multiplication
- Geometric linear transformations in 2 and 3-dimensional Spaces: Rotations, Reflections etc.
-Properties and Results

1, 2, 3, 4, 5, 6

Chapter: Linear Transformations:
Definition and Examples, Properties and results.

Tutorial 4

6

Chapter 6 Rank Nullity Theorem
- Nullspace and range;
- Rank Nullity Theorem

- Bijection & Isomorphisms; injective and surjection.

1, 2, 3, 4, 5, 6, 7

Chapter: and Rank Nullity Theorem

Isomorphisms, Injection
and Surjection

Tutorial 5

7

Chapter 7 Matrix Representation
-- for linear transformations between finite dimensional spaces.

- Standard Matrix

1, 2, 3, 4

Chapter: Matrix Representation

Tutorial 6

8

Chapter 7 (Continued)
- Matrix for Change of Basis/Coordinates

Chapter 8 Eigenvectors and Eigenvalues
- Definition and Examples
- for linear transformations and matrices, both real and complex spaces.

1, 2, 3, 4

Chapter: Matrix Representation of Linear transformations

Tutorial 7

9

Chapter 8 (Continued)
- Finding Eigenvectors and Eigenvalues
- Characteristic polynomials,
- Properties and Results.

Chapter 9
Diagonalization:
- Diagonal matrix
- Diagonalizable matrix

1, 2, 3, 4

Chapter: Change of basis, Matrix of Geometric linear transformations;
Eigenvectors and Eigenvalues

Tutorial 8

10

Chapter 9 (Continued) - Splitting Characteristics Polynomials

Chapter 10 Inner Product Spaces:
- Inner product and norm;

1, 2, 3, 4, 5, 6, 7, 8

Chapter: Eigenvectors and Eigenvalues

Tutorial 9

11

Chapter 10 (Continued)
- Some Inequalites: Cauchy Schwarz Inequality & Triangle Inequality

Chapter 11 Orthogonality
- Orthogonality
- Orthogonal complement
- Orthogonal projection.

1, 2, 3, 4, 5, 6, 7, 8, 9

Chapter: Diagonalization;
Inner Product and Norm.

Tutorial 10

12

Chapter 11 (Continued)
- Orthogonal and orthonormal basis.
- Gram-Schmidt Process
- Results

1, 2, 3, 4, 5, 6, 7, 8, 9

Chapter: Inner Product Space

Tutorial 11

13

Revision

1, 2, 3, 4, 5, 6, 7, 8, 9

Chapter: Inner Product Space

Tutorial 12

Appendix 1: Assessment Rubrics

Rubric for Lectures: Test 1 (15%)

Point-based Marking. Marks will be awarded to mathematical logical and correct justification or computation or examples. 

Rubric for Lectures: Test 2 (25%)

Point-based Marking. Marks will be awarded to mathematical logical and correct justification or computation or examples.

Rubric for Examination: Final Examination (60%)

Point-based Marking.