This lecture-based module continues to introduce students to the fundamental notions of linear algebra.  Picking up where MA1008 Linear Algebra I left off, we interpret bases as coordinate isomorphisms with kⁿ. We then explain how this allows us to write down and manipulate abstract linear maps as matrices. Matrix computation methods are introduced and studied at length. You will learn how to compute the kernel and the image of a linear map via matrices, and as an application, how to solve systems of homogeneous and non-homogeneous linear equations. Finally, we study how the matrix of a linear map changes with a change of basis and the standard form that it can be brought to.

The module concludes with some of the most exciting and challenging material in Linear Algebra.  We review the standard form that a matrix of a linear map can brought to by changing the bases of the source and the image space.  We then ask what happens in the case of linear maps from a vector space to itself, where we choose the same basis for the source and image space.  Diagonalisability is discussed, leading naturally to the notions of eigenvectors and eigenvalues. Students are taught how to compute these via the characteristic polynomial. We will also study dual vector spaces and dual bases, orthogonality, inner products, and the Gram-Schmidt orthonormalisation algorithm. This leads to the spectral theorem, which says that self-adjoint linear maps can be diagonalised.

Pre-requisite modules MA1005 and MA1008

• Understand the basic abstract notions of linear algebra: vector space, linear map, basis, etc.
• Understand and work with concrete examples of these abstract notions for the vector spaces of n-tuples of scalars, matrices, polynomials, etc.
• Understand matrices as a computational tool for working with linear maps and be able to work with them in that capacity
• Check a set of vectors for linear independence and/or spanning properties
• Solve homogeneous and non-homogeneous systems of linear equations
• Put the matrix of a linear map into its standard form
• Invert a matrix
• Compute the change-of-basis matrix and use it to transform matrices of linear maps 

You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 100 hours of self-guided study throughout the duration of the module, including preparation of formative assessments

Understand the basic abstract notions of linear algebra and be able to compute them in concrete numerical examples.

Transferable Skills:

Appreciation of abstract notions of linear algebra and their ubiquity in all branches of mathematics. Matrix manipulation: reduced row and echelon form, rank and nullity, inverting a matrix, determinant, change of basis. Solving systems of linear equations

THE OPPORTUNITY FOR REASSESSMENT IN THIS MODULE:

Opportunities for re-assessment is only permitted provided you have not failed more credit than in the resit rule adopted by your programme. If the amount of credit you have failed is more than permitted by the relevant resit rule, you may be permitted to repeat study if you are within the threshold set for the Repeat rule adopted by your programme. You will be notified of your eligibility to resit/repeat any modules after the Examining Board in the Summer period.

All resit assessments will be held in the Resit Examination period, prior to the start of the following academic session.

• Linear maps as matrices: a worked example. Flow chart of writing down linear maps as matrices.
• Bases as coordinate isomorphisms to kⁿ. Examples.
• Reduction to maps kⁿ→ k^m. Examples.
• The maps kⁿ→ k^m are the same as m × n matrices. Examples.
• Linear map composition as matrix multiplication. Examples.
• Row and column operations. Reduced echelon form.
• The rank and the nullity of a linear map. The row/column span, rank, and nullity of a matrix. Identifying these notions.
• The rank-nullity formula. The standard matrix of a linear map.
• Homogeneous and non-homogeneous systems of linear equations. Their associated matrices. Linear map interpretation.
• Solving a homogeneous system of linear equations by computing the kernel of its matrix.
• Solving an inhomogeneous system of linear equations by the extended matrix manipulation.
• Linear maps of vector spaces to themselves and their invertibility. Determinants. Invertibility of a matrix.
• Methods to invert a matrix.
• Change of basis and its associated matrix. Basis-independence of determinants.
• Diagonalisable linear maps. Examples.
• Eigenvectors, eigenvalues and eigenspaces. Diagonalisation as finding a basis of eigenvectors. Examples.
• Computing the eigenvalues and eigenvectors; characteristic polynomials.
• Non-diagonalisable linear maps. Geometric and algebraic multiplicities.
• Motivation: the usual dot product of kⁿ.
• Inner products over R and over C. Examples.
• Orthogonality. The orthogonal complement of a vector set.(U┴)┴=SpanU.
  Examples, e.g. (2 vectors)┴ = perpendicular line.
• Orthonormal bases. The Gram-Schmidt orthonormalisation. Examples.
• Diagonalisability of self-adjoint linear maps on inner product spaces. Specialisation to unitary, symmetric, orthogonal matrices. Examples.
• (Time permitting) The Jordan Normal Form of a linear map. The algorithm for computing it. Examples.