## Linear Algebra B

 Unit code: MATH10212 Credit Rating: 15 Unit level: Level 1 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

Co-Requisite
Prerequisite

#### 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.

#### 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.

#### 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.

#### Assessment methods

• Other - 25%
• Written exam - 75%

#### Assessment Further Information

Attendance at supervisions: weighting 5%

Submission of coursework at supervisions: weighting 5%

In-class test weighting 15%

Two and a half hour end of semester examination: weighting 75%

#### 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]

The course is based on the textbook:

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

#### 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.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Alexandre Borovik - Unit coordinator