MATH10212 - 2010/2011

 

Specification

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.

Brief Description of the unit

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

• be able to solve systems of linear equations by using Gaussian elimination to reduce the augmented matrix to row echelon form or to reduced row echelon form;
• understand the basic ideas of vector algebra: linear dependence and independence and spanning;
• be able to apply the basic techniques of matrix algebra, including finding the inverse of an invertible matrix using Gauss-Jordan elimination;
• know how to find the row space, column space and null space of a matrix, to find bases for these subspaces and be familiar with the concepts of dimension of a subspace and the rank and nullity of a matrix, and to understand the relationship of these concepts to associated systems of linear equations;
• be able to find the eigenvalues and eigenvectors of a square matrix using the characteristic polynomial and will know how to diagonalize a matrix when this is possible;
• be able to find the orthogonal complement of a subspace;
• be able to recognize and invert orthogonal matrices;
• be able to orthogonally diagonalize symmetric matrices;
• be familiar with the general notions of a vector space over a field and of a subspace, linear independence, dependence, spanning sets, basis and dimension of a general subspace;
• be able to find the change-of-basis matrix with respect to two bases of a vector space;
• be familiar with the notion of a linear transformation, its matrix with respect to bases of the domain and the codomain, its range and kernel, and its rank and nullity and the relationship between them;
• be familiar with the notion of a linear operator and be able to find the eigenvalues and eigenvectors of an operator.

Future topics requiring this course unit

Almost all Mathematics course units, particularly those in pure mathematics.

Syllabus

Linear Systems: Solving Linear Systems - Gauss' Method - Describing the Solution Set - General = Particular + Homogeneous. Reduced Echelon Form - Gauss-Jordan Reduction - Row Equivalence. [Hefferon, Chapter 1; 6 lectures]

Vector Spaces: Definition of Vector Space - Subspaces and Spanning Sets. Linear Independence - Definition and Examples. Basis and Dimension - Basis - Dimension Vector Spaces and Linear Systems. [Hefferon, Chapter 2; 6 lectures]

Maps Between Spaces: Isomorphisms - Definition and Examples - Dimension Characterizes Isomorphism. Homomorphisms - Definition - Rangespace and Nullspace. Computing Linear Maps -  Representing Linear Maps with Matrices - Any Matrix Represents a Linear Map. Matrix Operations - Sums and Scalar Products - Matrix Multiplication  - Mechanics of Matrix Multiplication - Inverses. Change of Basis - Changing Representations of Vectors - Changing Map Representations. Projection - Gram-Schmidt Orthogonalization - Orthonormal and orthogonal Matrices.  [Hefferon, Chapter 3; 9 llectures]  

Determinants: Definition - Properties of Determinants - Determinants Exist - Geometry of Determinants - Laplace's Expansion. [Hefferon, Chapter 4; 6 lectures]  

Similarity: Complex Vector Spaces - Factoring and Complex Numbers, a Review - Complex Representations. Similarity - Definition and Examples - Diagonalizability - Eigenvalues and Eigenvectors - Orthogonal Diagonalisation of symmetric matrices. [Hefferon, Chapter 5; 6 lectures]  

Textbooks

The course is based on the open source textbook:

• Jim Hefferon, Linear Algebra. Available for free download from ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf as well as from the course website, http://www.maths.manchester.ac.uk/~avb/math10212.html.

Some exercise problems will be borrowed from another open source textbook:

• Robert Beezer, A First Course in Linear Algebra.  Available for free download from http://linear.ups.edu/download.html as well as from the course website, http://www.maths.manchester.ac.uk/~avb/math10212.html.

Notes will be issued for the material not covered in the course text.

Teaching and learning methods

3 lectures and 1 example class per week

Assessment

Attendance at supervisions: weighting 5%

Submission of coursework at supervisions: weighting 5%

In-class test: weighting 15%

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

to the top

Arrangements