Page 18 - Matrices theory and applications

1
Elementary Theory

1.1 Basics

1.1.1 Vectors and Scalars

Fields. Let (K, +, ·)be a field.Itcould be IR, the field of real numbers, CC
(complex numbers), or, more rarely, QQ (rational numbers). Other choices
are possible, of course. The elements of K are called scalars.
Given a field k, one may build larger fields containing k: algebraic ex-
tensions k(α 1 ,... ,α n ), fields of rational fractions k(X 1 ,... ,X n ), fields of
formal power series k[[X 1 ,... ,X n ]]. Since they are rarely used in this book,
we do not define them and let the reader consult his or her favorite textbook
on abstract algebra.
The digits 0 and 1 have the usual meaning in a field K, with 0 + x =
1 · x = x. Let us consider the subring ZZ1, composed of all sums (possibly
empty) of the form ±(1 + ··· +1). Then ZZ1 is isomorphic to either ZZ or
toafield ZZ/pZZ. In the latter case, p is a prime number, and we call it the
characteristic of K. In the former case, K is said to have characteristic 0.
Vector spaces. Let (E, +) be a commutative group. Since E is usually
not a subset of K, it is an abuse of notation that we use + for the additive
laws of both E and K. Finally, let

(a, x) → ax,
K × E → E,