Department of Mathematical Sciences, Unit Catalogue 2011/12
|Level:||Intermediate (FHEQ level 5)|
|Supplementary Assessment:||MA20216 Mandatory extra work (where allowed by programme regulations)|
|Requisites:||Before taking this unit you must take MA10209 and take MA10210|
To deepen understanding of linear algebra, provide a solid introduction to inner product spaces, and introduce multilinear algebra, with emphasis on bilinear forms.
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.
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials)
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.
MA20216 is Compulsory on the following programmes:Department of Computer Science