Unit rationale, description and aim

Linear algebra is central to solving many real-world problems, including how search engines like Google rank web pages. More broadly, linear systems and vector spaces are essential in modelling networks, and simulating physical systems, making linear algebra vital for students in mathematics, engineering, and computer science. 

This unit builds upon the matrix and vectors topics in MATH107 to extend students’ knowledge of linear systems. Such systems arise naturally from geometry when finding the intersections of lines and planes, however such systems appear widely across mathematics and applied disciplines. Through the study of vector spaces and, linear transformations, students are introduced to concepts used in most other areas of mathematics.

This unit introduces vector spaces, dimension and linear transformations, which are used to define matrices, determinants, eigen-values and eigen-vectors. These ideas are developed through both theoretical exploration and practical examples to motivate their study. The unit also introduces the axiomatic approach, encouraging abstract reasoning and a deeper understanding of algebraic structures.

The aim of this unit is to introduce the concepts of linearity, dimension and related ideas. It seeks to provide students with the tools to analyse and solve complex linear problems and apply abstract thinking to both theoretical and applied settings

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2025 10

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Prerequisites

MATH107 Introduction to Logic and Algebra

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.

Explain basic theories and definitions of vector s...

Learning Outcome 01

Explain basic theories and definitions of vector space.
Relevant Graduate Capabilities: GC1, GC8

Determine whether a given set of vectors is linear...

Learning Outcome 02

Determine whether a given set of vectors is linearly dependent or independent.
Relevant Graduate Capabilities: GC1, GC8

Describe the concept of dimension of a vector spac...

Learning Outcome 03

Describe the concept of dimension of a vector space using bases.
Relevant Graduate Capabilities: GC1

Construct and interpret matrices and matrix multip...

Learning Outcome 04

Construct and interpret matrices and matrix multiplication by analysing linear transformations between vector spaces.
Relevant Graduate Capabilities: GC1

Evaluate the determinant of and find the eigen-val...

Learning Outcome 05

Evaluate the determinant of and find the eigen-values and eigen-vectors of a square matrix.
Relevant Graduate Capabilities: GC1, GC8

Apply Linear Algebra concepts and techniques to so...

Learning Outcome 06

Apply Linear Algebra concepts and techniques to solve relevant real-world problems.
Relevant Graduate Capabilities: GC1, GC7, GC8

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

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.

At specified times, through the semester, students will sit two supervised test components to ensure that students have fully integrated the learning and can independently bring a variety of strategies to bear to solve a variety of problems. The take home assignment will allow students to develop other skills including traditional and technological problem-solving methods.

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, the lecturer may request that assignments and tests are handwritten.

Overview of assessments

Assessment 1: Continuous assessment A single ta...

Assessment 1: Continuous assessment

A single task which is submitted in 1, 2 or 3 parts across the semester.

Weighting

30%

Learning Outcomes LO1, LO2, LO3, LO4, LO5, LO6
Graduate Capabilities GC1, GC7, GC8

Assessment 2: Test 1 Content and application te...

Assessment 2: Test 1

Content and application test – this test will be conducted under supervision and on campus.

Weighting

35%

Learning Outcomes LO1, LO2, LO3
Graduate Capabilities GC1, GC8

Assessment 3: Test 2 Content and application te...

Assessment 3: Test 2

Content and application test – this test will be conducted under supervision and on campus.

Weighting

35%

Learning Outcomes LO1, LO2, LO3, LO4, LO5, LO6
Graduate Capabilities GC1, GC7, GC8

At specified times, through the semester, students will sit two supervised test components to ensure that students have fully integrated the learning and can independently bring a variety of strategies to bear to solve a variety of problems. The take home assignment will allow students to develop other skills including traditional and technological problem-solving methods.

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 be delivered in a pre-recorded, asynchronous format structured around short, focused segments. These will typically provide explanations of the material to be covered, along with examples of it's applications, 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 by allowing students to pause, reflect, and consolidate their understanding before progressing to new material. Face-to-face tutorials then provide opportunities for students to develop and practice the skills introduced in lectures enabling them to build understanding, competence and confidence through guided problem-solving, discussion, and peer interaction. Continuous formative learning opportunities allow for the early detection of problems a student might be having and 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. In all cases this should be supported using available online technology.

Representative texts and references

Representative texts and references

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

Axler, S. (2015). Linear Algebra Done Right. 3rd Edition. New York: Springer

Beezer, R.A. (2012). A First Course in Linear Algebra. Version 3.00. Washington: Congruent Press

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

Lipschutz, S. & Lipson, M.L. (2012). Schaum's Outline of Linear Algebra. 5th Edition. New York: McGraw-Hill Education

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

Poole, D. (2014). Linear Algebra: A Modern Introduction. 4th Edition. Stamford: Cengage Learning

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

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

Locations
  • Brisbane
  • Canberra
  • Melbourne
  • Strathfield
Credit points
10
Year
2025

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