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Up: Year 2
Previous: MC240 Abstract Analysis
MC241 Linear Algebra
| Credits: 10 |
Convenor: Prof. G. Robinson |
Semester: 1 (weeks 7 to 12) |
| Prerequisites: |
essential: MC147 |
|
| Assessment: |
Regular coursework: 20% |
One and a half hour exam: 80% |
| Lectures: |
18 |
Classes: |
6 |
| Tutorials: |
6 |
Private Study: |
51 |
| Labs: |
none |
Seminars: |
none |
| Project: |
none |
Other: |
none |
| Total: |
75 |
|
|
Explanation of Pre-requisites
The student will be assumed to be familiar with the
general notion of a vector space, and to understand
the concepts of spanning set, basis, linear independence.
Course Description
The course continues the study of linear algebra, taking
a more conceptual point of view than MC147. There is a review
of some of the basic notions of vector spaces, linear independence,
spanning sets and bases. A detailed study
of linear transformations, their matrices and some of their major properties
then commences. A central theme is whether a vector space has
a basis consisting of eigenvectors for a given linear transformation,
or equivalently, whether it has a diagonal matrix with respect to a
suitable basis. The geometrically motivated notion of
inner product space is then studied, continuing with the study of
linear transformations which respect the inner product structure
of such a space.
Aims
To familiarize the
student with the central concepts of the subject.
To impart to the student an appreciation of the
importance of Mathematical structure and the
study of mappings which preserve that structure.
Objectives
To familiarize the student with the central Mathematical
notion of linear transformation, and with the
properties of such transformations. At the conclusion of the
course, the student should be able to decide whether a vector
space has a basis of eigenvectors for a given linear transformation
and appreciate how to choose a basis with respect to which
the matrix of the transformation has a particularly manageable form.
The student should also be able to construct an orthonormal
basis for a given finite-dimensional inner product space
and understand how to find an orthonormal basis of eigenvectors
of a normal linear transformation.
Transferable Skills
The course should assist the student in developing skills
in Mathematical writing and correct rigorous argument.
The underlying theory of this course has applications
in almost all later Mathematics modules. The introduction
to ideas of objects with Mathematical structure and mappings between
those objects which preserve aspects of their structure provided
by this course should provide preparation for several ensuing
Mathematics courses. In addition, several of the results of
the course find practical applications in a wide variety of
settings.
Syllabus
Review of definitions of field, vector space. Vector subspaces.
Linear independence, spanning sets, basis and dimension.
Finite-dimensional spaces. Direct sum decompositions.
Linear transformations. Kernel and image of a linear transformation.
Rnak, nullity and their relationship. The dual of a vector space.
The matrix of a linear transformation. Change of basis matrix.
Eigenvalues, eigenvectors,eigenspaces. Characteristic Polynomial
(including some discussion of the fundamental theorem of algebra
and working over the complex field). The Cayley-Hamilton theorem.
The minimum polynomial, and its relationship to the characteristic
polynomial. Relationship of the minimum
polynomial to the question of existence of basis of eigenvectors
and diagonalizability of matrices. Inner Product
Spaces ( real and complex), Gram-Schmidt Process, Unitary, Hermitian and
Normal transformations and the existence of an orthonormal
basis of eigenvectors for such transformations. Matrix interpretation
of this, and application to quadratic forms.
Reading list
Essential:
Recommended:
R .B. J. T. Allenby,
Linear Algebra,
Edward Arnold.
C.W. Curtis,
Linear Algebra, an Introductory Approach,
Springer.
J .B. Fraleigh and R. B. Beauregard,
Linear Algebra, 2nd edition,
Addison Wesley.
Details of Assessment
The coursework will consist of regularly assigned
exercise sheets. The January examination will contain three
questions.
Next: MC242 Introduction to Groups
Up: Year 2
Previous: MC240 Abstract Analysis
Roy L. Crole
10/22/1998