Introduction to Linear Algebra

Group

$(G,*)$ is a group iff $*$ is a binary operation on set $G$ satisfying:

• Non-empty
  $G \neq \emptyset$.
• Closed
  $\forall a,b \in G, a * b \in G$.
• Associateive
  $\forall a,b,c \in G, (a * b) * c = a * (b * c)$.
• Identity exists
  $\exists e \in G$ such that $\forall a \in G, e * a = a * e = a$.
• Inverse exists
  $\forall a \in G, \exists a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$.

Can either be finite or infinite.

Properties:

• Identity is unique.
• Inverse is unique for each element.
• $\forall a \in G, \bar{\bar{a}} = a$.

Examples:

• $(\mathbb{Z}, +)$
• $(\mathbb{R}, +)$
• $(\mathbb{R}\setminus{0}, \cdot)$
• Monster Group
  A finite group with roughly $8 \times 10^{53}$ elements.

Abelian Group

A group where the binary operation is commutative.

$(G,*)$ is abelian iff:

• $(G,*)$ is a group and
• $\forall a,b \in G, a * b = b * a$.

Ring

$(R,+,\cdot)$ is a ring iff:

• $(R,+)$ is an abelian group
• $\cdot$ is associative and distributive over addition
• $\cdot$ has a multiplicative identity element

Not studied in this module.

Field

$(F,+,*)$ is a field iff 2 operations $+$ and $\cdot$ on $F$ satisfying:

• $(F,+)$ is an abelian group
• $(F\setminus{0},\cdot)$ is an abelian group ($0$ is additive identity)
• $\forall a,b \in F, a\cdot b \in F$ ($1$ is multiplicative identity)
• multiplication is closed and distributive over addition
  $\forall a,b,c \in F, a\cdot(b+c) = a\cdot b + a\cdot c$

Additive identity ($0$) is excluded for multiplication ($*$) operatin because it doesn’t have an inverse.

Properties:

• $0$ and $1$ are unique.
• $F$ has at least 2 elements, $0$ and $1$.

Examples:

• $(\mathbb{R}, +, \cdot)$
• $(\set{0,1,2}, +\mod 3, \cdot\mod 3)$

Finite Fields

A field with a finite number of elements.

For a prime number $p$, $\mathbb{F}_p = (\set{0,1,2,\dots,p-1},+\mod p, \cdot\mod p)$ is a field.

Every finite field has size $p^k$ for some prime $p$ and an some integer $k$. And, for any prime $p$ and for any integer $k$, there exists a field with size $p^k$. BUT not every set with $p^k$ number of elements are a field.