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# BSc Actuarial Science and Mathematics / Course details

Year of entry: 2021

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## Course unit details:Linear Algebra B

Unit code MATH10212 15 Level 1 Semester 2 Department of Mathematics No

### Overview

This core course aims at introducing students to the fundamental concepts of linear algebra culminating in abstract vector spaces and linear transformations. The course starts with systems of linear equations and some basic concepts of the theory of vector spaces in the concrete setting of real linear n-space, Rn. The course then goes on to introduce abstract vector spaces over arbitrary fields and linear transformations, matrices, matrix algebra, similarity of matrices, eigenvalues and eigenvectors. The subject material is of vital importance in all fields of mathematics and in science in general.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Pure Mathematics B MATH10111 Pre-Requisite Compulsory
Calculus and Applications B MATH10232 Co-Requisite Compulsory

### Aims

This course unit aims to introduce the basic ideas and techniques of linear algebra for use in many other lecture courses. The course will also introduce some basic ideas of abstract algebra and techniques of proof which will be useful for future courses in pure mathematics.

### Learning outcomes

On successful completion of this course unit students will be able to:

• Use Gauss-Jordan elimination to solve systems of linear equations and to compute the inverse of an invertible matrix.
• Use the basic concepts of vector and matrix algebra, including linear dependence / independence, basis and dimension of a subspace, rank and nullity, for analysis of matrices and systems of linear equations.
• Evaluate determinants and use them to discriminate between invertible and non-invertible matrices.
• Use the characteristic polynomial to compute the eigenvalues and eigenvectors of a square matrix and use them to diagonalise matrices when this is possible; discriminate between diagonalisable and non-diagonalisable matrices.
• Orthogonally diagonalise symmetric matrices and quadratic forms.
• Combine methods of matrix algebra to compose the change-of-basis matrix with respect to two bases of a vector space.
• Identify linear transformations of finite dimensional vector spaces and compose their matrices in specific bases.

### Syllabus

Linear Equations in Linear Algebra: Systems of Linear Equations - Row Reduction and Echelon Forms - Vector Equations - The Matrix Equation Ax=b - Solution Sets of Linear Systems - Applications of Linear Systems - Linear Independence - Introduction to Linear Transformations - The Matrix of a Linear Transformation [Lay, Chapter 1, 6 lectures]

Matrix Algebra: Matrix Operations - The Inverse of a Matrix - Characterizations of Invertible Matrices - Partitioned Matrices - Matrix Factorizations - Subspaces of Rn - Dimensions and Rank [Lay, Chapter 2, 4 lectures]

Determinants: Introduction to Determinants - Properties of Determinants - Cramer's Rule, Volume, and Linear Transformations [Lay, Chapter 3, 4 lectures]

Vector Spaces: Vector Spaces and Subspaces - Null Spaces, Column Spaces, and Linear Transformations - Linearly Independent Sets; Bases - Coordinate Systems - The Dimension of Vector Space  Rank - Change of Basis [Lay, Chapter 4, 6 lectures]

Eigenvalues and Eigenvectors: Eigenvectors and Eigenvalues - The Characteristic Equation â€' Diagonalization - Eigenvectors and Linear Transformations - Complex Eigenvalues [Lay, Chapter 5, 6 lectures]

Orthogonality: Inner Product, Length, and Orthogonality - Orthogonal Sets - Orthogonal Projections - The Gram- Schmidt Process - Inner Product Spaces - Applications of Inner Product Spaces [Lay, Chapter 6, 4 lectures]

Symmetric Matrices: Diagonalization of Symmetric Matrices [Lay, Chapter 7, 2 lectures]

### Assessment methods

Method Weight
Other 25%
Written exam 75%

Attendance at supervisions: weighting 5%

Submission of coursework at supervisions: weighting 5%

In-class test weighting 15%

Examination: weighting 75%

### Feedback methods

Feedback supervisions will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

The course is based on the textbook:

• D. C. Lay, Linear Algebra and Its Applications, Pearson Education, 2010 (and previous editions).

### Study hours

Scheduled activity hours
Lectures 33
Tutorials 11
Independent study hours
Independent study 106

### Teaching staff

Staff member Role
Oliver Dorn Unit coordinator