ABSTRACT
In this chapter we begin with a review of elementary linear algebra, and in particular the geometry of Euclidean vector space Rn. The main purpose of this first section is to fix our conventions on notation and terminology. We then introduce the concept of a lattice, the main object of study throughout this book, and prove some basic lemmas about these structures. The last section of the chapter recalls some essential facts from the geometry of numbers, by which is meant the interplay between Euclidean geometry and the theory of numbers. Throughout this book we will use the following standard notation:
Z the domain of integers
Q the field of rational numbers
R the field of real numbers
C the field of complex numbers
Fp the field of congruence classes modulo the prime number p
We regard n-tuples of elements from a field F as either column vectors or as row vectors, and denote them by boldface roman letters:
x =
x1 x2 ... n
∈ Fn, x = [ x1, x2, · · · , xn ] ∈ Fn. 1
We use the column format when we consider an n×n matrix acting as a linear operator on Rn by left multiplication on column vectors. However, we will be primarily concerned with operations on a basis of Rn, and for this reason it is convenient to represent the basis vectors x1, x2, . . . , xn as the rows of an n×n matrix X . We can then represent operations on the basis as elementary row operations on the matrix. More generally, we can represent a general change of basis as left multiplication of X by an invertible n× n matrix C. Definition 1.1. For any field F, and any positive integer n, the vector space Fn consists of all n-tuples of elements from F, with the familiar operations of vector addition and scalar multiplication defined by
x+ y =
x1 x2 ... xn
+
y1 y2 ... yn
=
x1 + y1 x2 + y2
... xn + yn
, ax = a
x1 x2 ... xn
=
ax1 ax2 ...
