Linear Equations

1.1 Fields

This book assumes that the reader is familiar with elementary algebra of real and complex numbers.

Properties of Algebra

Let $F$ denote either the set of real numbers or the set of complex numbers.

1. Closure under Addition: For any two elements $a,b\in F$:

$$a+b\in F$$

2. Closure under Multiplication: For any two elements $a,b\in F$:

$$ab\in F$$

3. Commutativity of Addition: For any two elements $a,b\in F$:

$$a+b=b+a$$

4. Associativity of Addition: For any three elements $a,b,c\in F$:

$$(a+b)+c=a+(b+c)$$

5. Additive Identity: There exists an element $0\in F$ such that for any $a\in F$:

$$a+0=a$$

6. Additive Inverse: For each element $a\in F$, there exists an element $-a\in F$ such that:

$$a+(-a)=0$$

7. Commutativity of Multiplication: For any two elements $a,b\in F$:

$$ab=ba$$

8. Associativity of Multiplication: For any three elements $a,b,c\in F$:

$$(ab)c=a(bc)$$

9. Multiplicative Identity: There exists an element $1\in F$ such that for any $a\in F$:

$$a\cdot 1=a$$

10. Multiplicative Inverse: For each non-zero element $a\in F$, there exists an element $a^{-1}\in F$ such that:

$$a\cdot a^{-1}=1$$

Field

Suppose there is a set $F$ of elements $x$, $y$, $z$, … with two operations, addition and multiplication, that satisfy the properties 1-10 of algebra mentioned above. Then $F$ is called a field.

With the usual operations of addition and multiplication, the set $C$ of complex numbers is a field. The set $R$ of real numbers is also a field.

A subfield of the field $C$ is a set $F$ of complex numbers which is itself a field under the usual operations of addition and multiplication of complex numbers.