MST20050 Linear Algebra II

Academic Year 2022/2023

This module gives an introduction to the theory of (finite-dimensional) vector spaces over fields and linear transformations (i.e., the functions between vector spaces that preserve the vector space structure). The module relies on the theory of systems of linear equations and matrix algebra, developed in first year. The main topics are as follows: brief revision of Gaussian elimination and Gauss-Jordan elimination, vector spaces over a field, subspaces, spanning sets, linear independence, bases, dimension, coordinate spaces, matrix techniques, row space, column space, null space, rank and nullity of a matrix, eigenvalues, eigenvectors, diagonalization, revision of functions on sets and their properties, linear transformations, kernel and image, isomorphisms, matrix of a linear transformation, vector space of linear transformations, change of bases for matrices of linear transformations, rank and nullity of a linear transformation, quotient spaces, dimension formula, first isomorphism theorem, rank-nullity theorem.

[Disclaimer: module content and assessment strategies may be subject to minor changes during the trimester. These changes may not be reflected in this module descriptor at that time, but will be clearly communicated to all students via other means.]

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Curricular information is subject to change

Learning Outcomes:

After successful completion of this module, a student should be able to:
- Define and discuss the main concepts related to vector spaces and linear transformations.
- Construct examples of vector spaces, subspaces and linear transformations.
- Determine if a given set of vectors is linearly independent, a spanning set, a basis.
- Determine if a subset of a vector space is a subspace.
- Determine bases of the row space, column space and null space of a given matrix.
- Determine the rank and nullity of a given matrix.
- Compute the characteristic polynomial, eigenvalues and eigenvectors of a given matrix.
- Determine whether or not a matrix can be diagonalized.
- Compute with coordinatizations of vectors after a choice of basis.
- Compute the change of basis matrix for given bases of a vector space.
- Compute the matrix of a linear map with respect to a choice of bases.
- Determine kernel and image of a linear map.
- Determine if a linear map is injective, surjective, bijective.
- Determine rank and nullity of a linear map.


Indicative Module Content:

Brief revision of Gaussian elimination and Gauss-Jordan elimination, vector spaces over a field, subspaces, spanning sets, linear independence, bases, dimension, coordinate spaces, matrix techniques, row space, column space, null space, rank and nullity of a matrix, eigenvalues, eigenvectors, diagonalization, revision of functions on sets and their properties, linear transformations, kernel and image, isomorphisms, matrix of a linear transformation, vector space of linear transformations, change of bases for matrices of linear transformations, rank and nullity of a linear transformation, quotient spaces, dimension formula, first isomorphism theorem, rank-nullity theorem.

Student Effort Hours: 
Student Effort Type Hours
Lectures

24
Tutorial

12
Autonomous Student Learning

76
Total

112
Approaches to Teaching and Learning:
The intention is to have all contact hours face-to-face.

Some of the lectures may be in blended or on-line format. 
Requirements, Exclusions and Recommendations
Learning Requirements:

This course relies heavily on the theory of systems of linear equations. Anyone taking this course should have a good grasp of Gaussian elimination, Gauss-Jordan elimination, results relevant to solving systems of linear equations and basic matrix algebra including computing determinants and matrix inverses. A working knowledge of integer, rational, real and complex numbers and the differences between them is desirable. Students should have previously taken and passed MST10030 Linear Algebra I, MATH10200 Matrix Algebra or an equivalent course.


Module Requisites and Incompatibles
Pre-requisite:
MATH10200 - Matrix Algebra, MST10030 - Linear Algebra I

Incompatibles:
MATH10260 - Linear Algebra for Engineers, MATH20030 - Linear Algebra 2 (Sci)., MATH20300 - Linear Algebra 2 (MathSci)


 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Class Test: In term class test. Unspecified n/a Standard conversion grade scale 40% No

20
Examination: Final exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Ms Claire Mullen Tutor