Linear algebra (Osnabrück 2024-2025)/Part I/List of definitions

Definition:Empty set

The set which does not contain any element is called the empty set, denoted by

$\emptyset .$

Definition:Subset

Let ${}T$ and ${}M$ denote sets. ${}T$ is called a subset

of ${}M$ if every element of ${}T$ is also an element of ${}M$.

Definition:Intersection

For sets ${}L$ und ${}M$, we call

${}L\cap M={\left\{x\mid x\in L{\text{ and }}x\in M\right\}}\,$

the intersection

of the two sets.

Definition:Union

For sets ${}L$ und ${}M$, we call

${}L\cup M={\left\{x\mid x\in L{\text{ or }}x\in M\right\}}\,$

the union

of the sets.

Definition:Cartesian product

Suppose that two sets ${}L$ and ${}M$ are given. Then the set

${}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,$

is called the product set

(or Cartesian product) of the sets.

Definition:Mapping

Let ${}L$ and ${}M$ denote sets. A mapping ${}F$ from ${}L$ to ${}M$ is given by assigning, to every element of the set ${}L$, exactly one element of the set ${}M$. The unique element which is assigned to ${}x\in L$, is denoted by ${}F(x)$. For the mapping as a whole, we write

$F\colon L\longrightarrow M,x\longmapsto F(x).$

Definition:Injective

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called injective, if for two different elements ${}x,x'\in L$, also ${}F(x)$ and ${}F(x')$

are different.

Definition:Surjective

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called surjective, if for every ${}y\in M$, there exists at least one element ${}x\in L$, such that

${}F(x)=y\,.$

Definition:Bijective

Let ${}M$ and ${}L$ denote sets and suppose that

$F\colon M\longrightarrow L,x\longmapsto F(x),$

is a mapping. Then ${}F$ is called bijective if ${}F$ is injective as well as

surjective.

Definition:Inverse mapping

Let ${}F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping

$G\colon M\longrightarrow L$

which sends every element ${}y\in M$ to the uniquely determined element ${}x\in L$ with ${}F(x)=y$,

is called the inverse mapping of ${}F$.

Definition:Composition

Let ${}L,\,M$ and ${}N$ denote sets, let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappings. Then the mapping

$G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),$

is called the composition of the mappings

${}F$ and ${}G$.

Definition:Operation

An operation (or binary operation) ${}\circ$ on a set ${}M$ is a mapping

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.$

Definition:Field

A set ${}K$ is called a field if there are two binary operations (called addition and multiplication)

$+:K\times K\longrightarrow K{\text{ and }}\cdot :K\times K\longrightarrow K$

and two different elements ${}0,1\in K$, which fulfill the following properties.

1. Axioms for the addition:
   1. Law of associativity: ${}(a+b)+c=a+(b+c)$ holds for all ${}a,b,c\in K$.
   2. Law of commutativity: ${}a+b=b+a$ holds for all ${}a,b\in K$.
   3. ${}0$ is the neutral element of the addition, i.e. ${}a+0=a$ holds for all ${}a\in K$.
   4. Existence of the negative: For every ${}a\in K$, there exists an element ${}b\in K$ with ${}a+b=0$.
2. Axioms of the multiplication:
   1. Law of associativity: ${}(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds for all ${}a,b,c\in K$.
   2. Law of commutativity: ${}a\cdot b=b\cdot a$ holds for all ${}a,b\in K$.
   3. ${}1$ is the neutral element for the multiplication, i.e. ${}a\cdot 1=a$ holds for all ${}a\in K$.
   4. Existence of the inverse: For every ${}a\in K$ with ${}a\neq 0$, there exists an element ${}c\in K$ such that ${}a\cdot c=1$.
3. Law of distributivity: ${}a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ holds for all ${}a,b,c\in K$.

Definition:Factorial

For a natural number ${}n$, one puts

${}n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,$

and calls this ${}n$ factorial.

Definition:Binomial coefficient

Let ${}k$ and ${}n$ denote natural numbers with ${}k\leq n$. Then

${}{\binom {n}{k}}:={\frac {n!}{k!(n-k)!}}\,$

is called the binomial coefficient ${}n$ choose ${}k$

Definition:Polynomial in one variable

Let ${}K$ be a field. An expression of the form

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,$

with ${}a_{i}\in K$ and ${}n\in \mathbb {N}$,

is called a polynomial in one variable over ${}K$.

Definition:Degree of a polynomial

The degree of a nonzero polynomial

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$

with ${}a_{n}\neq 0$

is ${}n$.