Linear Algebra II

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Apply concepts and theorems from linear algebra to vector spaces other than Rn, in particular function spaces
• Explain the axiomatic structures of abstract linear algebra and apply them in simple proofs
• Find matrix representation of linear transformations on vectors spaces other than Rn.
• Determine whether a linear transformation given by a matrix is diagonalisable.

• Basic introduction to groups: Q, R and C under addition, Q*, R* and C* under multiplication, matrix groups, cyclic groups, permutation groups, sign of a permutation.
• Fields: R, Q, C, the field of two elements.
• Definition of a vector space over K (where K is a field).
• Examples of vector spaces including function spaces (functions from a set to K, differentiable functions, polynomials), subspaces.
• Linear independence, spanning sets (generalisation of Linear Algebra I).
• Basis and dimension.
• Linear transformations, examples including differentiation.
• Matrix representation of a linear transformation.
• Image and kernel of a linear map, dimension theorem.
• Isomorphism of vector spaces.
• Determinants (axiomatic description, properties).
• Eigenvalues, eigenvectors of linear transformations.
• Diagonalisation, diagonalisability.
• Cayley-Hamilton theorem.

Teaching and learning methods

Lectures, problem classes, workshops, private study

Resources & Reading list

General Resources

Any other book on Linear Algebra covering vector spaces other than R n can be used..

Textbooks

Robert Valenza (1993). Linear Algebra: An Introduction to Abstract Mathematics. Springer / Undergraduate Texts in Mathematics.

Anthony Martin and Harvey Michele (2012). Linear Algebra Concepts and Methods. Cambridge University Press.

Sheldon Axler (2015). Linear Algebra Done Right. Springer / Undergraduate Texts in Mathematics.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.

Repeat Information

Repeat type: Internal & External