## MATH202 Linear Algebra

Second Semester |

MATH202 is an introduction to the fundamental ideas and techniques of linear algebra, and the application of these ideas to computer science, the sciences and engineering.

### Paper details

The principal aim of this paper is for students to develop a working knowledge of the central ideas of linear algebra: vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors, the spectral theorem, and the applications of these ideas in science, computer science and engineering.

In particular, the course introduces students to one of the major themes of modern mathematics: classification of structures and objects. Using linear algebra as a model, we will investigate techniques that allow us to tell when two apparently different objects can be treated as if they were the same.

### Potential students

MATH 202 is compulsory for the Mathematics major and is of particular relevance also for students majoring in Statistics, Computer Science and Physics.

It is a prerequisite for MATH 301 (Hilbert spaces), MATH 304 (Partial Differential Equations), MATH 306 (Geometry of Curves and Surfaces) MATH 342 (Modern Algebra), MATH 361 (Numerical Analysis).

### Prerequisites

MATH 170

### Course Outline

The paper will cover the following topics:

- One-to-one and onto functions
- Basic group theory (definition, subgroups, group homomorphisms and isomorphisms)
- Vector spaces over the real and complex numbers (mainly finite-dimensional)
- Linear transformations and their properties, kernel and range, examples involving Euclidean spaces, spaces of polynomials, spaces of continuous functions
- Linear combinations, linear independence and spans, bases, dimension, standard bases, extending bases of subspaces, coordinate vectors and isomorphisms, rank-nullity theorem.
- Representation of linear transformations by matrices
- Diagonalisation, eigenvalues and eigenvectors
- Inner products, orthogonality, orthogonal projections, Cauchy-Schwartz inequality, inner-product norm, orthonormal bases, Gram-Schmidt process, orthogonal complements, minimisation problems
- Self-adjoint transformations and the spectral theorem, positive transformations and their square roots, Singular-value decomposition of a matrix.

### Required text

You should buy a copy of the Linear Algebra Outline Notes from the Printshop.

### Lecturer

TBA

### Office hours

TBA

To make an appointment outside of office hours, either talk to Astrid or Ilija after class or send an e-mail giving 4-5 times when you can come in.

### Lectures

3 hours per week, Monday, Wednesday, Friday 9-10.

### Tutorials

1 hour per week, starting in week 2. You will be streamed into one tutorial. In the tutorials, the tutor will work one or two problems on the board, and then answer questions.

### Internal Assessment

The internal assessment is made up of 50% from 4 assignments and 50% from two class tests. See the Course Information for the schedule.

### Terms

None

### Exam format

A combination of true/false, multiple choice, short-answer and long-answer questions; all questions to be answered.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(0.85E + 0.075A + 0.075T, 0.70E + 0.15A + 0.15T)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

and all quantities are expressed as percentages.

This means that the internal assessment counts for at least 15% and at most 30% of your final mark.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

### Network

In this network of roads from downtown Seattle, the unknown flow rates, in vehicles per day, can be solved using a system of 16 linear equations in the 17 unknowns. By hand this would be very tedious but by using MATLAB the task is simple.### Dynamical systems ...

The population of spotted owls can be predicted by the matrix modelwhich relates the number of juvenile, subadult and adult birds at time k+1 to the number at time k. Plotting the outcomes for years 1995 to 2020 gives the graph below from which it is evident the population will perish.

It turns out the parameter 0.18 in the matrix, giving the fraction of juveniles surviving to be subadults, is critical. By considering the eigenvalues and eigenvectors of the matrix we can decide how this must be changed so as to guarantee the survival of the owl population.

### Augustin Louis Cauchy

The French mathematician Augustin Louis Cauchy (1789-1857) proved a special case of the Cauchy-Schwarz Inequality, namely that given any 2n real numbers a

_{1}, a

_{2}, ..., a

_{n}, b

_{1}, b

_{2}, ..., b

_{n}, then we have the inequality

(a

_{1}b

_{1}+a

_{2}b

_{2}+ ... + a

_{n}b

_{n})

^{2}≤

(a

_{1}

^{2}+ a

_{2}

^{2}+ ... + a

_{n}

^{2}) x

(b

_{1}

^{2}+ b

_{2}

^{2}+ ... + b

_{n}

^{2}).

In MATH 202 this inequality is a special case of one established for vectors in a general inner product space. Apart from the inequality, there is a lunar crater named after Cauchy as well as a street in Paris (Rue Cauchy) and he is one of 72 prominent French scientists whose names are recorded on plaques on the Eiffel Tower.

### Diagonalization

The diagonalization methods described in MATH 202 enable us to simplify the equations of certain curves in 2-dimensional space. For example, the curve $x^2-xy+y^2=1$ becomes more easily recognised as an ellipse after a rotation of the $x$-$y$-axes anti-clockwise through 45° since its new equation becomes $x_1^2+3y_1^2=2$.Hover over or click on the picture above to see the new axes.