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## MA20216: Algebra 2A

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Intermediate (FHEQ level 5)
Period: Semester 1
Assessment Summary: EX 100%
Assessment Detail:
• Examination (EX 100%)
Supplementary Assessment: MA20216 Mandatory extra work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA10209 and take MA10210
Description: Aims:
To deepen understanding of linear algebra, provide a solid introduction to inner product spaces, and introduce multilinear algebra, with emphasis on bilinear forms.

Learning Outcomes:
After taking this unit, students should be able to:
* Demonstrate understanding of vector spaces and abstract linear algebra, including dual spaces and bilinear forms.
* State and prove basic results about inner product spaces and linear operators on such spaces.
* Diagonalise symmetric and hermitian matrices.
* Classify quadratic forms in theory and practice.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials)

Content:
Vector subspaces, sums and intersections, complementary subspaces, quotients. Dual spaces and bases, transpose of a linear map, annihilators.
Inner product spaces over R and C. Cauchy-Schwarz inequality and applications.
Gram-Schmidt orthonormalization. Orthogonal subspaces and complements. Duality.
Linear operators on inner product spaces. Orthogonal and unitary groups; (skew) symmetric and (skew) hermitian matrices. Properties of eigenvalues and eigenspaces.
Finite dimensional spectral theorem. Applications.
Bilinear forms, relation with dual spaces, nondegeneracy. Tensor products and applications. Multilinear forms, alternating forms, determinants revisited.
Alternating bilinear forms, classification. Quadratic forms, relation to symmetric bilinear forms. Sylvester's law and classification over R and C. Applications.
Programme availability:

#### MA20216 is Compulsory on the following programmes:

Department of Computer Science
• USCM-AFB20 : BSc(Hons) Computer Science and Mathematics (Year 2)
• USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 2)
• USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 2)
• USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 2)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 2)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 2)
Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 2)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 2)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 2)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 2)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 2)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 2)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 2)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 2)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 2)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 2)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 2)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 2)
• USMA-AFB05 : BSc(Hons) Statistics (Year 2)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 2)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 2)
Department of Physics
• USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 2)
• USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 2)
• USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 2)
• USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 2)
• USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 2)
• USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 2)

Notes:
* This unit catalogue is applicable for the 2014/15 academic year only. Students continuing their studies into 2015/16 and beyond should not assume that this unit will be available in future years in the format displayed here for 2014/15.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.