Page 21 - Matrices theory and applications

1. Elementary Theory
4
CC
is a CC-vector space, with
One verifies easily that G
CC
dim CC G
Furthermore, G may be identified with an IR-linear subspace of G
x → (x, 0).
= G + iG. In a more general setting,
Under this identification, one has G CC =dim IR G. CC by
one may consider two fields K and L with K ⊂ L, instead of IR and CC,but
L
the construction of G is more delicate and involves the notion of tensor
product. We shall not use it in this book.
One says that a polynomial P ∈ L[X] splits over L if it can be written
as a product of the form
r

a (X − a i ) , a, a i ∈ L, r ∈ IN, n i ∈ IN .
∗
n i
i=1
Such a factorization is unique, up to the order of the factors. A field L in
which every nonconstant polynomial P ∈ L[X] admits a root, or equiva-
lently in which every polynomial P ∈ L[X] splits, is algebraically closed.If
the field K contains the field K and if every polynomial P ∈ K[X]admits

arootin K , then the set of roots in K of polynomials in K[X]is an alge-

braically closed field that contains K, and it is the smallest such field. One
calls K the algebraic closure of K.Every field K admits an algebraic clo-

sure, unique up to isomorphism, denoted by K. The fundamental theorem
of algebra asserts that IR = CC. The algebraic closure of QQ, for instance,
is the set of algebraic complex numbers, meaning that they are roots of
polynomials P ∈ ZZ[X].
1.1.2 Matrices
Let K be a field. If n, m ≥ 1, a matrix of size n × m with entries in K is a
map from {1,... ,n}× {1,... ,m} with values in K. One represents it as
an array with n rows and m columns, an element of K (an entry)at each
point of intersection of a row an a column. In general, if M is thenameof
the matrix, one denotes by m ij the element at the intersection of the ith
row and the jth column. One has therefore
 
m 11 ... m 1m
. .
. . .  ,
 . 
. . .
M = 
m n1 ... m nm
which one also writes
M =(m ij ) 1≤i≤n,1≤j≤m .
In particular circumstances (extraction of matrices or minors, for example)
the rows and the columns can be numbered in a different way, using non-