Advanced Algebra

Author: Zhang Xianke / et al.
Editor-in-Chief: Qiu Jingnan
Publisher:
Publish Date: 1998-03-01
Features:
Content Summary
This book primarily covers linear algebra, including numbers and polynomials, determinants, systems of linear equations, matrices, linear spaces, quadratic forms, linear transformations, space decomposition, matrix similarity, Euclidean spaces and unitary spaces, bilinear forms, tensor products, and exterior products. The content is profound, making it suitable for building a strong foundation; the perspective is modern, facilitating adaptation to contemporary mathematics. There are also several advanced supplementary topics. It can serve as a textbook for mathematics or computer science majors in universities or as a reference for other disciplines. The book was written based on the authors' long-term teaching experience in linear algebra at the University of Science and Technology of China and Tsinghua University, as well as their research in algebra. It references several renowned international textbooks. The book includes a wealth of examples and exercises of varying difficulty, with answers and hints provided at the end, along with appendices, a bilingual name index, and a bibliography.
Excerpt:
In fact, from \( a^{-1} = e \cdot a^{-1} = (a^{-1} \cdot a) \cdot a^{-1} = a^{-1} \cdot (a \cdot a^{-1}) \), multiplying both sides on the left by \( (a^{-1})^{-1} \) yields \( e = a \cdot a^{-1} \). It is also clear that \( a \cdot e = a \cdot (a^{-1} \cdot a) = (a \cdot a^{-1}) \cdot a = e \cdot a \). If the group \( (G, ) \) also satisfies the commutative property, i.e., \( a \cdot b = b \cdot a \) for all \( a, b \in G \), then the group is called an Abelian group or a commutative group. The operation of an Abelian group is often denoted by addition (using \( + \) instead of \( \) as the operator), the identity element is often denoted as \( 0 \) (called the zero element), and the inverse of \( a \) is often denoted as \( -a \) (called the negative of \( a \)). Examples of Abelian groups include \( (\mathbb{Z}, +) \), \( (\mathbb{Q}, +) \), \( (\mathbb{R}, +) \), and \( (\mathbb{C}, +) \), where the addition \( + \) refers to ordinary number addition.
Definition 1.2
A ring (ring) is a set \( R \) in which two binary operations, denoted by addition (\( + \)) and multiplication (\( \cdot \)), are defined, and it satisfies:
(1) \( (R, +) \) is an Abelian group;
(2) \( (R, \cdot) \) is a semigroup, meaning it satisfies closure and associativity;
(3) The distributive law \( a \cdot (b + c) = a \cdot b + a \cdot c \) and \( (a + b) \cdot c = a \cdot c + b \cdot c \) hold for all \( a, b \in R \).
Such a ring is denoted as \( (R, +, \cdot) \) or simply \( R \), and the multiplication symbol \( \cdot \) is often omitted, writing \( a \cdot b \) as \( ab \). The additive identity is often denoted as \( 0 \).
Note that \( 0 \cdot a = a \cdot 0 = 0 \) for all \( a \in R \). In fact, \( 0a = (0 + 0)a = 0a + 0a \), which implies \( 0a = 0 \).
If the ring \( R \) has a multiplicative identity \( e \), then \( R \) is called a unital ring. In a unital ring \( R \), for \( c \in R \), if there exists \( x \in R \) such that \( xc = cx = e \), then \( x \) is called the inverse of \( c \), and \( c \) is said to be invertible (or \( c \) is a unit of \( R \)). If the multiplication in a ring \( R \) satisfies the commutative property, then \( R \) is called a commutative ring.
Definition 1.3
A field (field) is a ring \( (F, +, \cdot) \) such that the set of all nonzero elements \( F \) forms an Abelian group under multiplication. In other words, a field is a set \( F \) with two binary operations (\( + \) and \( \cdot \)) that satisfies:
(1) \( (F, +) \) is an Abelian group;
(2) \( (F, \cdot) \) is an Abelian group;
(3) The distributive law.
Examples:
1.2 \( (\mathbb{Z}, +, \cdot) \) is a ring called the ring of integers, which is a very important ring (here the operations are ordinary addition and multiplication).
1.3 \( \mathbb{Q} \), \( \mathbb{R} \), and \( \mathbb{C} \) are fields under their usual addition and multiplication, respectively, called the field of rational numbers, the field of real numbers, and the field of complex numbers. These are commonly used and very important fields.
1.4 \( \mathbb{Q}(\sqrt{2}) = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \} \) is a field.
If a subset \( K \) of a field \( F \) is itself a field under the original operations of \( F \), then \( K \) is called a subfield of \( F \), and \( F \) is called an extension field of \( K \). Similar definitions exist for subgroups and subrings. Subfields of the complex field \( \mathbb{C} \) are called number fields. The three fields in the above examples are number fields. There are many number fields (infinitely many), and they are important fields.
Note that every number field always contains the natural number \( 1 \), and thus contains \( \mathbb{Z} \), and thus contains \( \mathbb{Q} \). Therefore, the field of rational numbers \( \mathbb{Q} \) is the smallest number field and is a subfield of any number field. There are also many fields other than number fields (infinitely many), and they are very important. The following example is a "binary field," which is very important in information coding:
1.5 \( \mathbb{F}_2 = \{ 0, 1 \} \) is a field under the following-defined addition and multiplication:
\( 0 + 0 = 0 \), \( 0 + 1 = 1 + 0 = 1 \).
Hereafter, \( 0 \) and \( 1 \) will be used to denote the additive and multiplicative identities of a field \( F \), respectively.
In advanced linear algebra, it is common to study functions, polynomials, vectors, etc., over a field \( F \). In earlier elementary textbooks, the base field \( F \) is often taken to be the real number field \( \mathbb{R} \). Most of the discussions in this book are carried out over a general base field \( F \) to meet the theoretical needs of further mathematical development and practical application requirements in areas such as computer information and communication. For a general field \( F \), we often refer to its elements as numbers (although they are not necessarily complex or real numbers), in contrast to polynomials and vectors over \( F \).
1.2 Congruence and Congruence Classes in Integers
An important property of the ring of integers \( \mathbb{Z} \) is that it supports division with remainder, i.e., if \( m, n \in \mathbb{Z} \) and \( m \neq 0 \), then there exist \( q, r \in \mathbb{Z} \) such that \( n = mq + r \) and \( 0 \leq r < |m| \); here \( q \) is called the quotient of \( n \) divided by \( m \), and \( r \) is called the remainder. If \( r = 0 \), then \( m \) divides \( n \), denoted as \( m \mid n \). From the division with remainder property of \( \mathbb{Z} \), many other properties of \( \mathbb{Z} \) can be derived, such as the fundamental theorem of arithmetic (i.e., every integer can be uniquely factored into a product of primes, which will be proven in Section 1.6, and this section discusses congruences of integers using this property).
If integers \( a \) and \( b \) have the same remainder when divided by \( m \), then \( a \) and \( b \) are said to be congruent modulo \( m \), denoted as \( a \equiv b \pmod{m} \), which is equivalent to \( ma \equiv mb \pmod{m} \), and also equivalent to \( a = b + mk \) for some \( k \in \mathbb{Z} \). The symbol \( \equiv \) is called the congruence symbol, and it is read as "congruent to." The above expression is called a congruence.
Congruence has the following analogous properties to equality (for all \( a, b, c, d \in \mathbb{Z} \)):
1. (Transitivity) If \( a \equiv b \pmod{m} \) and \( b \equiv c \pmod{m} \), then \( a \equiv c \pmod{m} \).
2. (Symmetry) If \( a \equiv b \pmod{m} \), then \( b \equiv a \pmod{m} \).
3. (Reflexivity) \( a \equiv a \pmod{m} \) always holds.
4. (Addition of congruences) If \( a \equiv b \pmod{m} \) and \( c \equiv d \pmod{m} \), then \( a + c \equiv b + d \pmod{m} \).
5. (Multiplication of congruences) If \( a \equiv b \pmod{m} \) and \( c \equiv d \pmod{m} \), then \( ac \equiv bd \pmod{m} \).
6. (Reduction of congruences)
(1) If \( a \equiv b \pmod{m} \), and \( d \mid a \), \( d \mid b \), and \( d \) is coprime to \( m \), then \( a \equiv b \pmod{m} \).
(2) If \( a \equiv b \pmod{m} \) and \( d \) is a common divisor of \( a \), \( b \), and \( m \), then \( a/d \equiv b/d \pmod{m/d} \).
The concept of congruence was first introduced by Gauss and is of great significance. The modulus \( m \) is usually taken to be a positive integer.
Example 1.6 (Modular 9 Test) Let the sum of the digits of the decimal representation of a positive integer \( a \) be divided by 9, and the remainder is denoted as \( a \). For example, \( 72982 = 1 \). The modular 9 test asserts that if \( a \times b = c \), then \( a \times b \equiv c \pmod{9} \); similarly, if \( a + b = c \), then \( a + b \equiv c \pmod{9} \). This can be used to initially check the correctness of calculations. For example, for \( 72982^2 = 5326372334 \), since the right-hand side modulo 9 is 2, the equality is incorrect. To prove the modular 9 test, note that \( 10 \equiv 1 \pmod{9} \). Therefore, if the decimal representation of \( a \) is \( a = a_n10^n + \cdots + a_110 + a_0 \), then \( a \equiv a_n + \cdots + a_1 + a_0 \equiv a \pmod{9} \). Thus, if \( ab = c \), then \( ab \equiv c \pmod{9} \), which is the modular 9 test.