Vector Spaces

Vector Space

Let $F$ be a field. A $F$-vector space is a set $V$ with two operations:

• Addition: $+:V\times V\to V,(v_1,v_2)\mapsto v_1+v_2$
• Scalar multiplication: $\cdot :F\times V\to V,(c,v)\mapsto c\cdot v$

which satisfy the following properties:

• Associativity: $\forall v_1,v_2,v_3\in V,a,b\in F,(v_1+v_2)+v_3=v_1+(v_2+v_3),(a\cdot b)\cdot v_1=a\cdot (b\cdot v_1)$
• Commutativity: $\forall v_1,v_2\in V,v_1+v_2=v_2+v_1$
• Additive Identity: $\exists 0\in V,\forall v\in V,0+v=v$
• Additive Inverse: $\forall v\in V,\exists w\in V,v+w=0$
• Multiplicative Identity: $\exists 1\in F,\forall v\in V,1\cdot v=v$
• Distributivity: $\forall a,b\in F,v_1,v_2\in V,a\cdot (v_1+v_2)=a\cdot v_1+a\cdot v_2,(a+b)\cdot v_1=a\cdot v_1+b\cdot v_2$

e.g. The set \newcommand{\R}{\mathbb{R}}F[a,b]=\{f\colon[a,b]\to\R\} is a vector space on $\mathbb{R}$ under the addition and scalar multiplication defined by $$(f+g)(x)=f(x)+g(x),\quad (\alpha f)(x) = \alpha f(x)$$$C[a,b]={f\colon[a,b]\to\R\mid f \text{ is continuous}}$ is a subspace under the same addition and scalar multiplication.

Proposition

Let $V$ be an $F$-vector space, $v\in V,c\in F$, then

• $0_F\cdot v=0_V$
• $c\cdot 0_V=0_V$
• $(-1)\cdot v=-v$
• $c\cdot v=0⟹c=0_F\text{ or }v=0_V$

Proof $0_F\cdot v=(0_F+0_F)\cdot v=0_F\cdot v+0_F\cdot v$. It follows that $$0=0_{F}\cdot v+ (-0_{F}\cdot v)=0_{F}\cdot v$$$\square$

Subspace

A subspace $W$ of a vector space $V$ is a subset $W\subseteq V$ which is also a vector space inherit the same operations as in $V$.

Proposition

A subset $W\subseteq V$ is a subspace of a $F$-vector space $V$ if and only if

• $0\in W$
• $w_1,w_2\in W⟹w_1+w_2\in W$
• $c\in F,w\in W⟹cw\in W$

Span and Linear Combinations

Span

The span of a set of vectors $S\subseteq V$ is the set of all finite linear combinations of vectors in $S$: $\mathrm{span}(S)=\left\{{\sum }_{i=1}^n{c}_i{v}_i\mid n\in \mathbb{N},v_i\in S,c_i\in F\right\}.$

Proposition

The span of any nonempty set of vectors is a subspace.