Linear Algebra -- Dr Stoy -- 14 MT + 8 HT

Linear algebra pervades and is fundamental to geometry 
(from which it originally arose), algebra, 
analysis, applied mathematics, statistics--indeed
all of mathematics.

The course has several aims.  
The first is to introduce students through a thorough study of 
two- and three-dimensional spaces to the general concept of 
a vector space, subspaces, and the ideas of 
linear dependence, independence, spanning sets, bases, dimension. 

A second aim is to introduce students to matrices and their 
applications to the algorithmic solution of 
systems of linear equations   
and to the study of linear transformations of vector spaces. 

A third aim is to introduce determinants and their properties.  
A fourth aim is to introduce eigenvalue theory and 
some of its applications. 

Fourteen lectures in Michaelmas Term

Introduction: examples of linear problems 
(e.g., system of linear equations, differential equations) 
and their solutions.  
Vectors in the plane and 3-space, and co-ordinates. 
Addition of vectors and multiplication of vectors by 
scalars corresponding to co-ordinatewise operations 
in R^2 and R^3.

Linear combinations of vectors in R^2 and R^3.
Lines and planes as subspaces of R^2, R^3.
The subspace spanned by a set of vectors.

Linear independence of vectors in R^2, R^3.  
Bases; co-ordinates with respect to a basis; co-ordinate changes.  

Linear transformations of R^2,R^3, 
with geometric examples.  
Specification of such transformations by matrices.  
Examples of simplification by good choice of basis.  
Kernels and images.

Definition of a vector space 
(over R; brief mention of C and Q); 
examples (e.g. R^n, polynomials).  
Simple consequences of the axioms.  
Subspaces; examples.

The span of a (finite) set of vectors; spanning sets; examples.  
Finite dimensionality 
[all spaces should be finite-dimensional for the rest of the course].

Linear dependence and linear independence. 
Definition of a basis; co-ordinates with respect to a basis.

Sums and intersections of subspaces; forumla for the dimensions of the sum.
Reduction of a spanning set and extentions of a linearly indpendant 
set to a basis; proof that all bases have the same size. Dimension of a space.

Linear transformations from one (real) vector space to another; 
examples. Rectangular matrices over R (brief mention of C and Q); 
row and column vectors as matrices. For given bases, correspondence between 
linear transformations and matrices. Sums, scalar multiples.  
Composition of transformations and product of matrices.

The image and kernel of a linear transformation.  
The rank-nullity theorem. Applications.

Elementary row operations on matrices; 
echelon form and row-reduction.  
Invariance of the row space under row operations; 
row rank.  Applications to finding bases.

Matrix representation of a system of linear equations.  
Significance of image, kernel, rank and nullity 
for systems of linear equations.   
Solution by Gaussian elimination.  
Bases of solution space of homogeneous equations.  

Invertible matrices; use of row operations to decide invertibility and 
to calculate inverse. Column space and column rank.  
Equality of row rank and column rank.

Eight lectures in Hilary Term

The matrix of a linear transformation with respect to bases, 
and change of bases.

Determinants of square matrices
[facts about permutations and parity to be stated and used as 
necessary--proofs to be given later in the term];
properties of the determinant function; determinants and the scalar 
triple product.  Computation of determinant by reduction to row echelon form. 
Proof that a square matrix is invertible if and only if it is non-singular.
Determinant of a linear transformation of a vector space to itself.

Eigenvalues of linear transformations of a vector space 
to itself.  The characteristic polynomial of a square matrix; 
the characteristic polynomial of a linear transformation 
of a vector space to itself.  The linear independence of a set of 
eigenvectors associated with distinct eigenvalues; 
diagonalisability of matrices.

Reading List

C. W. Curtis, Linear Algebra - An Introductory Approach, 
Springer (4th edition), reprinted 1994

R. B. J. T.  Allenby, Linear Algebra, Arnold, 1995

T. S.  Blyth and E. F.  Robertson, Basic Linear Algebra,  Springer, 1998

D. A.  Towers, A Guide to Linear Algebra, Macmillan, 1988

D. T.  Finkbeiner, Elements of Linear Algebra, Freeman, 1972 
[Out of print, but available in many libraries]

B. Seymour Lipschutz, Marc Lipson, Linear Algebra Third Edition 2001