# Linear Algebra Summer School MATHS2031

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 2 (SCQF level 8)**Typically Offered:**Summer**Available to Visiting Students:**Yes**Available to Erasmus Students:**No

#### Short Description

This course introduces many basic techniques of Linear Algebra that are applicable in the

physical and chemical sciences, statistics and more or less every branch of mathematics. The two

central topics are: the basic theory of vector spaces; and the concept of a linear transformation,

with emphasis on the use of matrices to represent linear maps.

#### Timetable

Running over an eight-week period parallel with a separate course on Differential Equations,

the courses are divided into two four-week blocks with two-thirds of the material of this course

being covered in the first and one-third of the material being covered in the second (for the

Differential Equations course the opposite is true). In total there will be approximately 29 contact hours comprising lectures, problem sessions/tutorials and guided learning.

#### Requirements of Entry

Normally a student should have completed at least one semester of algebra, covering solving

simple systems of linear equations and the basic theory of vectors.

#### Excluded Courses

None

#### Co-requisites

Students must be enrolled in both the Linear Algebra Summer School and Scotland, the City of Glasgow and the Origins of the Modern World courses

#### Assessment

Description of Summative Assessment:

Assessment: Unseen examination (50%) 90 minutes exam paper consisting of 6 short written questions. Course work consists of handwritten assignments (20%) and online assignments (20%); a further 5% will come from participation (as opposed to grades) in weekly quizzes and other assigned work, with a final 5% coming from oral assessment and presentation, including group work.

Reassessment: In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

**Main Assessment In:** August

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non-Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

Because of the timing of the summer school, there is no opportunity for reassessment to be timetabled within the same academic session.

#### Course Aims

This course introduces many basic techniques of Linear Algebra that are applicable in the

physical and chemical sciences, statistics and more-or-less every branch of mathematics. The two

central topics are: the basic theory of vector spaces; and the concept of a linear transformation,

with emphasis on the use of matrices to represent linear maps. Eigenvalues and eigenvectors are

also defined and studied so that the process of diagonalising a square matrix can be discussed.

The emphasis is on achieving understanding of the concepts and results and learning to use

them.

The course will use a variety of traditional and computerised assessment methods to provide

regular feedback to both the students and tutors in order to both empower students to gauge

their progress and to customise the course to the their needs.

#### Intended Learning Outcomes of Course

Students should be familiar with all definitions and results covered in the course, should understand the proofs of results, and should be able to apply the results to problems. Students should also learn to be rigorous and logical in their presentation of solutions.

By the end of the course, students should be able to

_ perform matrix arithmetic

_ recognise vector spaces and subspaces over R and C;

_ test sets of vectors for linear independence and spanning properties, and understand methods for obtaining bases for a specified subspace of a vector space;

_ decide whether or not a map between spaces is linear, describe a linear map in matrix form, and calculate various objects (eg image, kernel) associated with a linear map;

_ evaluate determinants recursively and using elementary row and column operations, factorise algebraic determinants, and apply results about determinants in theoretical problems;

_find the characteristic polynomial of a square matrix, and use it to determine the eigenvalues and eigenvectors of the matrix;

_ use the eigenvalues and eigenvectors of a matrix to diagonalise it (where possible) using a suitable invertible matrix;

_ recognise an orthonormal set of vectors and find an orthonormal basis for a given subspace of Rn.

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment. The final grade for the course will

be comprised of 50% final exam and 50% continuous assessment. Students will be required to

have a passing grade in both halves of the course assessment in order to secure an overall pass.