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Subsections
* 3.2.1.1 Synopsis
* 3.2.1.2 Reading List
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3.2.1 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.
3.2.1.1 Synopsis
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
[$\mathbb{R}^2$] and [$\mathbb{R}^3$].
Linear combinations of vectors in [$\mathbb{R}^2$] and [$\mathbb{R}^3$].
Lines and planes as subspaces of [$\mathbb{R}^2$], [$\mathbb{R}^3$]. The
subspace spanned by a set of vectors.
Linear independence of vectors in [$\mathbb{R}^2$], [$\mathbb{R}^3$]. Bases;
co-ordinates with respect to a basis; co-ordinate changes. Testing for
linear independence (introducing and using [$2 \times 2$] and [$3 \times 3$]
matrices).
Linear transformations of [$\mathbb{R}^2$], [$\mathbb{R}^3$], 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 [$\mathbb{R}$]; brief mention of
[$\mathbb{C}$] and [$\mathbb{Q}$]); examples (e.g., [$\mathbb{R}^n$],
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.
Linear transformations from one (real) vector space to another; examples.
Rectangular matrices over [$\mathbb{R}$] (brief mention of [$\mathbb{C}$]
and [$\mathbb{Q}$]); 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.
Reduction of a spanning set and extension of a linearly independent set to a
basis; proof that all bases have the same size. Dimension of a space.
Elementary row operations on matrices; echelon form and row-reduction.
Invariance of the row space under row operations; row rank. Applications to
finding bases.
Sums and intersections of subspaces; formula for the dimension of the sum.
The image and kernel of a linear transformation. The rank-nullity theorem.
Applications.
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.
3.2.1.2 Reading List
1. C. W. Curtis, Linear Algebra--an Introductory Approach, Springer (4th
edition), reprinted 1994
2. R. B. J. T. Allenby, Linear Algebra, Arnold, 1995
3. T. S. Blyth and E. F. Robertson, Basic Linear Algebra, Springer, 1998
4. D. A. Towers, A Guide to Linear Algebra, Macmillan, 1988
5. D. T. Finkbeiner, Elements of Linear Algebra, Freeman, 1972 [Out of
print, but available in many libraries]
6. B. Seymour Lipschutz, Marc Lipson, Linear Algebra Third Edition 2001
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