## 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

If:

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

all expressed out of 100, then your final mark F in this paper will be calculated according to this formula:

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

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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### 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.