Algebra Notes 1

1.1) Fundamental theorem of algebra (D'Alembert): 

Every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ complex roots. 

For more details see here

1.2) Number field definition (A):

Let $F$ be a subset of $\mathbb{C}$ which contains at least two elements. We say that $F$ is a number field if the following conditions are met: 

a) if $z_1$ and $z_2$ are arbitrary numbers from $F$, then the numbers $z_1 + z_2$, $z_1 - z_2$, $z_1 \cdot z_2$ are also in $F$, 

b) if $z_1$ and $z_2$ are arbitrary numbers from $F$ and $z_2 \ne 0$, then $z_1 / z_2$ is also in $F$. 

1.3) Number field definition (B):

Let $F$ be a subset of $\mathbb{C}$ which contains at least two elements. We say that $F$ is a number field if the following conditions are met: 

a) if $z_1$ and $z_2$ are arbitrary numbers from $F$, then the numbers $z_1 + z_2$, $z_1 - z_2$, $z_1 \cdot z_2$ are also from $F$; 

b) if $z \in F$ and $z \ne 0$, then $z^{-1} \in F$ 

1.4) Note: Not all fields in mathematics are considered number fields (e.g. finite Galois fields are not considered number fields). The number sets $\mathbb{Q}, \mathbb{R}, \mathbb{C}$ (equipped with the usual algebraic operations) are number fields. The number sets $\mathbb{N}, \mathbb{Z}$ are not number fields.

1.5) Theorem: Every number field is a superset of the field of the rational numbers $\mathbb{Q}$.

1.6) Notations: 

a) $F_{m \times n}$ - the set of all $m \times n$ matrices with elements from the number field $F$

b) $M_{n}(F)$ - the set of all $n \times n$ square matrices with elements from the number field $F$; these are also called square matrices of order $n$ 

c) If $A = (a_{ij})_{m\times n}$ is a matrix, then by $A^t$ we denote the transposed matrix of the matrix $A$; it is defined as follows: $A^t = (a_{ji})_{n\times m}$ 

d) $E_n$ or just $E$ is the identity matrix ($E_n$ just emphasizes the fact that the order of the matrix is $n$)

e) $E_{ij}$ is the matrix from $F_{m \times n}$ which has element $1$ at position $(i,j)$ and zero elements at all other positions $(k,l) \ne (i,j)$

1.7) Operations on matrices:

a) sum of matrices: $A+B$

b) multiplication of a matrix with a number: $\lambda A$

c) transposed matrix: $A^t$

1.8) Property:

Obviously if $A = (a_{ij})_{m\times n}$ is an arbitrary matrix from $F_{m \times n}$ then $$A = \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \cdot E_{ij}$$

1.9) Definition:  

a) The matrices which have a single row (or a single column) are called row matrices (or column matrices), or n-tuples, or also n-dimensional vectors.

b) The set of all n-tuples with elements from the number field $F$ is denoted by $F^n$ 

c) The n-tuples  

$e_1 = (1, 0, 0, \dots, 0, 0),\ e_2 = (0, 1, 0, \dots, 0, 0),\ e_3 = (0, 0, 1, \dots, 0, 0),\ \dots\ ,\ $ 

$e_{n-1} = (0, 0, 0, \dots, 1, 0),\ e_{n} = (0, 0, 0, \dots, 0, 1)$ 

are called unit vectors or unit n-dimensional vectors.

1.10) Note: If $a = (a_1, a_2, \dots, a_n) \in F^n$ then obviously $a = a_1 e_1 + a_2 e_2 + \dots + a_n e_n$ This expression for $a$ is called a linear combination. In this particular case, $a$ is linear combination of the unit vectors $e_1, e_2, \dots, e_n$ 

1.11) Properties of the transposed matrix:

a) $(A^t)^t = A$ 

b) $(A+B)^t = A^t + B^t$ 

c) $(\lambda A)^t = \lambda A^t$ (for any $\lambda \in F$)