MATH220 - Linear Algebra

Year

Credit points

Campus offering

Prerequisites

MATH107 Introduction to Logic and Algebra

Incompatible

MATH212

Unit rationale, description and aim

This unit builds upon the matrix and vectors topics in MATH107 to extend students’ knowledge of linear systems and to solve important real world problems. Linear systems arise naturally from geometry when finding the intersections of lines and planes, however such systems appear quite generally throughout mathematics, science and in many real world applications. Approached through the study of vector spaces and, most importantly linear transformations, linear algebra introduces concepts that are used in most other areas of mathematics.

This unit introduces vector spaces, dimension and linear transformations. These ideas will be used to define matrices, determinants, eigen-values and eigen-vectors. Important examples will be used to motivate the study of linear algebra.

The aim of this unit is to introduce the concepts of linearity, dimension and related ideas. At the same time, students will gain some experience with the axiomatic approach to defining algebraic objects and how those abstract constructs can have important applications in the world.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Use the definition of a vector space to prove some simple results (GA5, GA9)

LO2 - Decide whether a given set of vectors is linearly dependent or independent (GA5, GA10)

LO3 - Define the dimension of a vector space using bases (GA5, GA9, GA10)

LO4 - Use the idea of linear transformations between vector spaces to define matrices and matrix multiplication (GA5, GA9)

LO5 - Evaluate the determinant of and find the eigen-values and eigen-vectors of a square matrix (GA5, GA10)

LO6 - Apply their knowledge of Linear Algebra to solve relevant real-world problems (GA5, GA9, GA10)

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA9 - demonstrate effective communication in oral and written English language and visual media 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics will include:

1. Revision of vectors, linear dependence and independence. Geometrical applications
2. Properties of vectors. Vector spaces and important examples
3. Vector subspaces. Linear transformations
4. Bases
5. Coordinate vectors and dimension of a vector space
6. Revision of matrices
7. Linear transformations and matrices
8. Determinants
9. Eigenvalues, eigenvectors and diagonalisation
10. Applications of eigenvalues

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

 This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

 150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems in linear algebra. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students.

The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

Overview of assessments

Representative texts and references

Anton, H. & Kaul, A (2019). Elementary Linear Algebra, 12th Edition New York: John Wiley & Sons

Kirkwood, J.R. & Kirkwood, B.H. (2020). Linear Algebra, Chapman Hall

Lay, D.C. & Lay, S.R. (2014). Linear Algebra and Its Applications. 5th Edition New York:Pearson Education

Penney, R.C. (2020). Linear Algebra: Ideas and Applications. 5th Edition New York: John Wiley & Sons

Strang, G. (2016) Introduction to Linear Algebra. 5th Edition Wellesley:Wellesley-Cambridge

Strang, G. (2020) Linear Algebra and Its Applications. Kindle Edition.