1.  Introduction:  This second semester course follows on from Linear Algebra I which is a prerequisite. This unit is a prerequisite for many second year units for mathematics students. The course looks at vectors and matrices from a more sophisticated viewpoint by introducing the concepts of vector spaces and linear transformations.

2.  Hours, Credits and Assessment:  The course is a 10 UCU unit of 20 lectures. Support teaching is via 6 seminars (shared with Real Analysis). Assessment is by homework (20%) at fortnightly intervals and examination (80%).

3.  Overview:  In Linear Algebra I we introduced matrix multiplication as a way of writing a system of linear equations as a single matrix equation. A more sophisticated viewpoint is to see multiplication by an m-by-n matrix as describing a function from n-dimensional to m-dimensional space. To understand better what this function is doing it is usually necessary to make a change of coordinates in the various spaces. To formalise this properly, we introduce the notion of an abstract vector space, and linear transformations between vector spaces. The key concept for vector spaces is that of a basis (--what we might informally have called a coordinate system), and we show that any two bases of a vector space have the same number of elements (the dimension of the vector space). We distinguish certain subsets of a vector space which can themselves be considered as vector spaces (subspaces), and relate the dimensions of subspaces to the dimension of the ambient space.

We then develop some general results about linear transformations. A linear transformation is determined by its effect on a basis, and, given bases, we can encode all the information needed to describe a linear transformation in a matrix. Changing the bases changes the matrix, and we describe this precisely. We also investigate how ‘simple’ this matrix can be made by the right choice of bases. This leads to the notions of eigenvalues and eigenvectors of (square) matrices, and the final section of the course is devoted to a study of these, how to compute them and applications.

The origins of linear algebra are in the middle of the 18th century when mathematicians began to analyse systems of (linear) equations more closely. The theory of determinants, already known to Leibniz in 1693, and much cultivated in the 19th century also provided an important stimulus. The notion of vector space is due to Grassmann (1844) and the vector space axioms in use today are almost exactly the same as those developed by Peano (1888). These more modern viewpoints on an old theory have applications everywhere in mathematics and wherever mathematics is applied. They provide unity and clarity in such diverse areas as differential equations, geometry, analysis, algebra, number theory and discrete mathematics. The axiomatisation leads one to consider vector spaces over fields other than the real numbers, for example over finite fields. This has enormously important applications in digital technologies via areas such as error correcting codes.

A O Morris Linear Algebra, (Van Nostrand)

H Anton Elementary Linear Algebra, (Wiley)

Fields: The g.c.d. of 2 integers and the Euclidean algorithm. Modular arithmetic; Fields. (2 lectures)

Motivation for vector spaces: Matrices as functions, examples of change of basis. (1 lecture)

Vector spaces: Definition of a vector space. Examples (main example Rn; function spaces). Linear combinations and subspaces. Linear independence. Definition of a basis and existence of dimension. Dimensions of subspaces and the modular law. Further examples. (6 lectures)

 Linear Transformations: Definition and examples (the linear transformation arising from a matrix; examples in analysis; geometric examples such as rotation and projection). Kernel and image of a linear transformation. Existence and uniqueness of linear transformations with prescribed images on a basis. The rank+nullity theorem. Examples (projections, derivatives and integrals of polynomials, linear equations) and applications. (4 lectures)

The matrix of a linear transformation once bases are chosen. Examples. Change of basis formulas for matrices. (2 lectures)

Eigenvalues and eigenvectors: The characteristic polynomial. Eigenspaces. Diagonalising matrices. Connection with eigenvectors. Examples of non-diagonalisability. Proof that 'n distinct eigenvalues' implies 'A diagonalisable'. Examples. Computing powers and roots of diagonalisable matrices. Inner product and orthogonal vectors in Rn . The Gram-Schmidt process. Diagonalisability of real symmetric matrices. Examples. (5 lectures)