Definitions

Def. 1

A field $F$ is a set together with two laws of composition:

  • addition: $F \times F \stackrel{+}{\rightarrow} F \quad (a, b \rightarrow a b)$

  • multiplication: $F \times F \stackrel{\times}{\rightarrow} F \quad (a, b \rightarrow a+b)$

that satisfies 3 axioms:

  1. Addition makes $F$ into an abelian group $F^{+}$ with identity element 0.
  2. Multiplication is commutative, and it makes the set of nonzero elements of $F$ into an abelian group $F^{\times}$ with identity element 1.
  3. distributive law: For all $a, b$, and $c$ in $F, a(b+c)=a b+a c$.

Notice, the axiom 1,2 describe the properties of $\times$ and $+$. Axiom 3 relates axioms 1,2.

Def. 2

A subfield of the field $\mathbb{C}$ (complex numbers) is a subset that is *closed* under four operations $+, -, \times, \div$ and contains $1$.

i.e. $F$ is a subfield of $\mathbb{C}$ if it has the following properties:

\[\begin{array}{ll} (+)\quad a,b \in F &\Rightarrow a+b \in F\\ (-)\quad a \in F &\Rightarrow -a \in F\\ (\times)\quad a,b \in F &\Rightarrow ab \in F\\ (\div)\quad a \in F(a\neq 0) &\Rightarrow a^{-1} \in F\\ (1)\quad 1\in F \end{array}\]

The above implies

  1. $0\in F$
  2. $F$ is a subgroup of additive group $\mathbb{C^+}$
  3. $F \backslash {0}$ is a subgroup of the multiplicative group $\mathbb{C^\times}$

Def.3

A vector space $V$ over a field $F$ is a set together with two laws of composition:

  • addition: $V \times V \rightarrow V\quad (v, w \rightarrow v+w$ for $v,w \in V)$

  • scalar multiplication by elements of the field:

    $F \times V \rightarrow V\quad (c, v \rightarrow cv$ for $c \in F, v\in V)$

that satisfies 4 axioms:

  1. Addition makes $V$ into a commutative group $V^+$ with identity element $0$.
  2. $1v=v$ for all $v\in V$.
  3. associative law: $(ab)v=a(bv)$, for all $a,b \in F$ and $v\in V$.
  4. distributive laws: $(a+b)v=av+bv$ and $a(v+w)=av+aw$, for all $a,b \in F$ and $v,w \in V$.

Def.4

A subset $W$ of $\mathbb{R^n}$ is a subspace if

  1. $w,w’\in W\Rightarrow w+w’\in W$.
  2. $c\in \mathbb{R},w \in W \Rightarrow cw\in W.$
  3. zero vector $\in W$

This is equivalent of saying:

  1. $W$ is non-empty.

  2. $c_1,c_2,…,c_n\in \mathbb{R}, w_1,w_2,…,w_n\in W$

    $\Rightarrow$ linear combination $c_1w_1+…+c_nw_n \in W$