Module Catalogues, Xi'an Jiaotong-Liverpool University   
Module Code: MTH107
Module Title: Advanced Linear Algebra
Module Level: Level 1
Module Credits: 5.00
Academic Year: 2020/21
Semester: SEM1
Originating Department: Mathematical Sciences
Pre-requisites: MTH007 OR MTH015 OR MTH017
This module serves as a second course in linear algebra. We present the general concepts and theory of linear spaces. We also introduce powerful tools in linear algebra for applications in science and engineering and introduce students to one of the major themes of modern mathematics: the classification of mathematical objects and structures.

After completion of the module, students should be well prepared for further study of topics such as abstract algebra, numerical analysis, scientific computing and multivariable statistics.

Learning outcomes 
A. Recognise vector spaces and linear transformations between them.

B. Interpret linear transformations of R^2 and R^3 in geometrical terms.

C. Determine the structure of the set of solutions of linear equations.

D. Convert a linear transformation from one representation to another.

E. Solve problems using the theory and techniques of inner product spaces.

F. Work with linear transformations to solve problems and to prove simple results.

Method of teaching and learning 
This module is delivered through formal lectures and tutorial classes
General vector spaces: fields, linear independence, span, basis, dimension, coordinates, subspaces, sum and direct sum.

General linear transformations: kernel, range, rank and nullity, dimension theorem, invariant subspaces, composition, inverse. Isomorphism, change of basis, similarity, eigenvalues and eigenvectors. Geometry of linear transformations in R^2 and R^3. Solution structure of linear equations.

Inner product spaces: inner product, norm, orthogonal basis, Gram-Schmidt process, orthogonal matrices, coordinates relative to an orthogonal basis, QR decomposition, orthogonal subspaces, orthogonal projection and decomposition, fundamental theorem of linear algebra.

Spectral theory of matrices: diagonalizability of matrices, eigenspaces, minimal polynomial, nilpotent matrices, Jordan form, Cayley Hamilton theorem, spectral decompositions: Jordan decomposition and applications.

Bilinear forms and quadratic forms, positive quadratic forms and matrices, characterization of positive matrices, invariant properties of congruent matrices.
Delivery Hours  
Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total
Hours/Semester 39     13      98  150 


Sequence Method % of Final Mark
1 Midterm 15.00
2 Final Examination 75.00
3 Assignments 10.00

Module Catalogue generated from SITS CUT-OFF: 6/5/2020 8:20:35 PM