# M1: Linear Algebra I - Material for the year 2018-2019

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14 lectures

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 number systems. The course begins with examples focussed on $\mathbb{R}^2$ and $\mathbb{R}^3$, 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.

Students will:

(i) understand the notions of a vector space, a subspace, linear dependence and independence, spanning sets and bases within the familiar setting of $\mathbb{R}^2$ and $\mathbb{R}^3$;

(ii) 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 prove results related to these concepts;

(iii) 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.

See the examinable syllabus.

Systems of linear equations.

Matrices and the beginnings of matrix algebra.

Use of matrices to describe systems of linear equations.

Elementary Row Operations (EROs) on matrices.

Reduction of matrices to echelon form.

Application to the solution of systems of linear equations.

Inverse of a square matrix.

Row reduced echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method.

Transpose of a matrix;

orthogonal matrices.

Vector spaces: definition of a vector space over a field (such as $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{C}$).

Subspaces.

Many explicit examples of vector spaces and subspaces.

Span of a set of vectors.

Examples such as row space and column space of a matrix.

Linear dependence and independence.

Bases of vector spaces; examples.

The Steinitz Exchange Lemma; dimension.

Application to matrices: row space and column space, row rank and column rank.

Coordinates associated with a basis of a vector space.

Use of EROs to find bases of subspaces.

Sums and intersections of subspaces; the dimension formula.

Direct sums of subspaces.

Linear transformations: definition and examples (including projections associated with direct-sum decompositions).

Some algebra of linear transformations; inverses.

Kernel and image, Rank-Nullity Theorem.

Applications including algebraic characterisation of projections (as idempotent linear transformations).

Matrix of a linear transformation with respect to bases.

Change of Bases Theorem.

Applications including proof that row rank and column rank of a matrix are equal.

Bilinear forms; real inner product spaces; examples.

Mention of complex inner product spaces.

Cauchy--Schwarz inequality.

Distance and angle.

The importance of orthogonal matrices.

All of the following books are held both in the Radcliffe Science Library (RSL) and in many (indeed, most) College libraries.

(1) Gilbert Strang, *Introduction to linear algebra* (Fifth edition, Wellesley-Cambridge 2016). http://math.mit.edu/~gs/linearalgebra/

(2) T.S. Blyth and E.F. Robertson, *Basic linear algebra* (Springer, London, 1998).

(3) Richard Kaye and Robert Wilson, *Linear algebra* (OUP, Oxford 1998), Chapters 1-5 and 8.

[More advanced but useful on bilinear forms and inner product spaces.]

(4) Charles W. Curtis, *Linear algebra - an introductory approach* (Springer, London, Fourth edition, reprinted 1994).

(5) R. B. J. T. Allenby, *Linear algebra* (Arnold, London, 1995).

(6) D. A. Towers, *A guide to linear algebra* (Macmillan, Basingstoke, 1988).

(7) Seymour Lipschutz and Marc Lipson, *Schaum's outline of linear algebra* (McGraw Hill, New York & London, Fifth edition, 2013).