MST20050 Linear Algebra II

Academic Year 2020/2021

This module gives an introduction to the theory of vector spaces, with an emphasis on finite dimensional spaces. The approach taken here is to give the student an intuition or feel for the geometry behind the numerical computations often associated with Linear Algebra. The module relies on the theory of systems of linear equations and matrix algebra. The main topics taught in the course are as follows: vector spaces over a field, subspaces, spanning sets, linear independence, bases, dimension, linear transformations and matrices, isomorphism, rank and nullity of a linear transformation, the Rank-Nullity Theorem, the rank of a matrix, eigenvalues and eigenvectors, diagonalising a matrix, inner-product spaces, orthonormal bases, and time allowing, positive-definite matrices.

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Curricular information is subject to change

Learning Outcomes:

After successful completion of this module, a student should be able to:
1. Define and discuss the basic concepts of linear independence and span.
2. Determine when a subset is a subspace and when a given map is a linear transformation.
3. State several basic results on vector spaces relating bases, subspaces and dimension;
4. Compute the characteristic polynomial, eigenvalues and eigenvectors of a matrix and to determine whether or not a matrix can be diagonalized.
5. Apply the rank-nullity theorem and state the connection between linear transformations and matrices.
6. Apply Gaussian elimination techniques to:
(a) decide whether or not a given set of vectors is linear independent;
(b) decide whether a vector is in the span of a set of vectors;
(c) decide whether or not a vector is contained in the column space or row space or null space of a matrix;
(d) compute a basis for the column space, row space or null space of a matrix;
(e) compute a basis for the eigenspace of a matrix for each eigenvalue;
(f) obtain a matrix representation of a linear transformation.
7. Evaluate inner products on a vector space.
8. Compute an orthogonal/orthonormal basis using the Gram-Schmidt process.
9. Visualise and make sense of learning outcomes 1-8.

Indicative Module Content:

Module indicative contentis as follows: vector spaces over a field, subspaces, spanning sets, linear independence, bases, dimension, linear transformations and matrices, isomorphism, rank and nullity of a linear transformation/matrix, the Rank-Nullity Theorem, eigenvalues and eigenvectors, diagonalising a matrix, inner-product spaces, orthonormal bases, and and time allowing, positive definite matrices.

Student Effort Hours: 
Student Effort Type Hours
Lectures

24
Tutorial

12
Autonomous Student Learning

76
Total

112
Approaches to Teaching and Learning:
All lectures and tutorials will be on-line. 
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 the Gauss-Jordan algorithm, results relevant to solving systems of linear equations and basic matrix algebra including how to compute the inverse of an nxn matrix for n<5. 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)

Additional Information:
This module builds on ideas from Linear Algebra 1 & Matrix Algebra, so students need to be proficient with solving systems of linear equations using Gaussian Elimination to decide if solutions are (a) a unique, (b) infinitely many, or (c) none.


 
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

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