Synopsis for Linear Algebra I

Overview

Linear algebra pervades and is fundamental to algebra, geometry, analysis, applied mathematics, statistics, and indeed most of mathematics. This course lays the foundations, concentrating mainly on vector spaces and matrices over the real and complex numbers. The course begins with examples focussed on and , and gradually becomes more abstract. The course also introduces the idea of an inner product, with which angle and distance can be introduced into a vector space.

Learning Outcomes

Students will:

1. understand the notions of a vector space, a subspace, linear dependence and independence, spanning sets and bases within the familiar setting of and ;
2. understand and be able to use the abstract notions of a general vector space, a subspace, linear dependence and independence, spanning sets and bases and be able to formally prove results related to these concepts;
3. have an understanding of matrices and of their applications to the algorithmic solution of systems of linear equations and to their representation of linear maps between vector spaces.

Synopsis

Systems of linear equations. Expression as an augmented matrix (just understood as an array at this point). Elementary Row Operations (EROs). Solutions by row reduction.

Abstract vector spaces: Definition of a vector space over a field (expected examples ). Examples of vector spaces: solution space of homogeneous system of equations and differential equations; function spaces; polynomials; as an -vector space; sequence spaces. Subspaces, spanning sets and spans. (Emphasis on concrete examples, with deduction of properties from axioms set as problems).

Linear independence, definition of a basis, examples. Steinitz exchange lemma, and definition of dimension. Coordinates associated with a basis. Algorithms involving finding a basis of a subspace with EROs.

Sums, intersections and direct sums of subspaces. Dimension formula.

Linear transformations: definition and examples including projections. Kernel and image, rank nullity formula.

Algebra of linear transformations. Inverses. Matrix of a linear transformation with respect to a basis. Algebra of matrices. Transformation of a matrix under change of basis. Determining an inverse with EROs. Column space, column rank.

Bilinear forms. Positive definite symmetric bilinear forms. Inner Product Spaces. Examples: with dot product, function spaces. Comment on (positive definite) Hermitian forms. Cauchy-Schwarz inequality. Distance and angle. Transpose of a matrix. Orthogonal matrices.

Reading

1. T. S.  Blyth and E. F.  Robertson, Basic Linear Algebra (Springer, London, 1998).
2. R. Kaye and R. Wilson, Linear Algebra (OUP, 1998), Chapters 1-5 and 8. [More advanced but useful on bilinear forms and inner product spaces.]

Alternative and Further Reading

1. C. W. Curtis, Linear Algebra – An Introductory Approach (Springer, London, 4th edition, reprinted 1994).
2. R. B. J. T.  Allenby, Linear Algebra (Arnold, London, 1995).
3. D. A.  Towers, A Guide to Linear Algebra (Macmillan, Basingstoke, 1988).
4. D. T.  Finkbeiner, Elements of Linear Algebra (Freeman, London, 1972). [Out of print, but available in many libraries.]
5. B. Seymour Lipschutz, Marc Lipson, Linear Algebra (McGraw Hill, London, Third Edition, 2001).