Source of most content: Gunhee Cho

Group

Def. Given a set $S$, we call $* : S\times S \rightarrow S$ a binary operation if $* $ satisfies associatvie law, $\forall x,y,z \in S, \ (x * y) * z = x * (y * z) $, and we call $(S, *) $ the semi group.

Def. Given a semi group $(M, *)$, if $\exists e \in M$ s.t. $\forall a \in M, \ a * e = e * a = a$, we call $e$ identity of $M$, and call $(M, * )$ the monoid.

NB1; Given a monoid, identity $e$ is unique.

Def. Given a monoid $(G, * )$, we call $G$ a group if $G$ satisfies $\forall x \in G, \exists y \in G$ s.t. $ x * y =y * x = e$.

NB2; Given a group, for each $a \in G$, inverse element $x \in G$, $a*x=x *a=e$, is unique.

Def. Given a group $(G, *)$, we call $(G, * )$ a commutative group if $\forall x,y \in G, \ x * y = y * x$.

NB3; If a binary operation is addtion $( * := + )$, then we call $(G,+)$ abelian group.

Ring and Field

Def. A set $R$ is a ring if $R$ is equipped with the addition $+:R\times R$ and multiplication $\cdot:R\times R$ satsifying:

1. $(R, +)$ is abelian group.
2. $(R, \cdot)$ is semi-group.
3. $+, \ \cdot$ is comparanle, i.e., $\forall x,y,z \in R, \ (x+y)\cdot z = x \cdot z + y \cdot z$ and $x \cdot (y+z) = x \cdot y + x \cdot z$.

Def. A ring $(F, +, \cdot)$ is called a field if $(F-\{ 0_F \}, \cdot)$ is a commutative group, where $0_F$ is identity of operation $+$.

NB4; $(\mathbb{R}, +, \times)$ is a field.

Def. Given a ring $R$, a set $M$ is called $R$-module if $M$ is equipped with addition $+$ and scalar multiplication, $R\times M \rightarrow M$ $(r, m) \mapsto r\cdot m$, where $r$ is scalar, satisfying:

1. $(M, +)$ is abelian group.
2. $\forall r_1, r_2 \in R, \forall m \in M, \ (r_1+r_2)\cdot m = r_1 \cdot m + r_2 \cdot m$.
3. $\forall r \in R, \forall m_1, m_2 \in M, \ r \cdot (m_1 +m_2) = r\cdot m_1 + r \cdot m_2$.
4. $\forall m \in M, \ 0_R \cdot m = 0_M$, where $0_R$ and $0_M$ are the identity of $(R, +),\ (M, +)$ respectively.
5. $\forall r_1, r_2 \in R, \forall m \in M, \ (r_1 r_2) \cdot m = r_1 \cdot (r_2 m)$.

Vector space

Def. Given a ring $R$ and a set $A$, we call $A$ an algebra if $A$ has 3 operations with compatibility:

1. $+ : A \times A \rightarrow A$.
2. $\times: A \times A \rightarrow A$.
3. $\cdot: R \times A \rightarrow A$.

Def. Given a field $F$, we call $V$ a vector space over $F$ if $V$ is $F$-module.

More precisely, following the definition of module;

Def. Given a set $V$ with a field $F$, we call $V$ a vector space over $F$ $(V/F)$ if following holds:

1. $(V,+)$ is abelian group.
2. scalar multiplication $F\times V \rightarrow V \ ((a,v)\mapsto a \cdot v = av)$ requires the compatibilities:
   • $\forall a_1, a_2 \in F, \forall v \in V, \ (a_1 + a_2)\cdot v = a_1 \cdot v + a_2 \cdot v$.
   • $\forall a \in F, \forall v_1, v_2 \in V, \ a \cdot (v_1 + v_2) = a\cdot v_1 + a\cdot v_2$.
   • $\forall v \in V, \ 1_{F}\cdot v = v, \ 0_F \cdot v = 0_V$.

Def. Given a vector space $V/F$, for a subset $S \subseteq V$:

1. span$(S)= $<$S$>$ := \{ a_1 v_1 + \cdots + a_n v_n = \sum^n_{k=1} a_k v_k \mid v_i \in S, a \in F \} \subseteq V$.
2. $S \subseteq V$ is linearly independent if for any $a_i \in F, v_i \in V$ satisfy $a_1 v_1 + \cdots + a_n v_n = 0_V$, then $a_i = 0_F \ \forall i = 1, \ldots n$.

NB5; span$(S)$ contains vectors that can be expressed as the finite summation.
NB6; Take $S = V$, then <$S$>$ = $<$V$> $ = V$, i.e., $\exists S \subseteq V$ s.t. <$S$>$ = V$.
NB7; Let $S \subseteq V$ be a linearly dependent subset. Thus, $\exists a_1, \cdots, a_n \in F$, not all zeros s.t. $a_1 v_1 + \cdots + a_n v_n = 0_V$ for some $v_1, \cdots, v_n \in V$. Assume $a_1 \ne 0$, then $a_1 v_1 = -( a_2 v_2 + \cdots a_n v_n) \Leftrightarrow v_1$ can be expressed by linear combination of $v_2, \cdots, v_n$.

Def. Given a vector space $V/F$, we call $\beta \subseteq V$ a basis of $V$ if:

1. $\beta$ is linearly independent.
2. span$(\beta)=V$.

NB8; If $\beta = \{ a \}, a \in V - \{ 0 \} $, then $\beta$ is linearly independent, since if $ka=0$ then $k=0$.