Vector Spaces, Subspaces and Span

Vector space: collection of vectors satisfying specific axioms.

Real and Complex Vector Spaces

A vector space has four key components:

1. Vectors
2. scalars
3. Vector addition
4. scalar multiplication

Definition: A vector space is a set $V$ together with a scalar field ($F=\mathbb{R}$ or $\mathbb{C}$), such that two operations are possible:

1. vector addition: for any $u,v\in V$, $u+v\in V$
2. scalar multiplication: for any $c\in F$ and $v\in V$, $cv\in V$

Operations must satisfy the following axioms for all $u,v,w\in V$ and scalars $a,b\in F$:

• Associativity of addition: $(u+v)+w=u+(v+w)$
• commutativity of addition: $u+v=v+u$
• Additive identity: $\exists 0\in V$ s.t. $v+0=v$
• Additive inverse: $\forall v\in V$, $\exists -v\in V$ s.t. $v+(-v)=0$
• Distributivity of scalar multiplication over vector addition: $a(u+v)=au+av$
• Distributivity over scalar addition: $(a+b)v=av+bv$
• Compatibility of scalar multiplication: $(ab)v=a(bv)$
• Multiplicative identity: $\exists 1\in F$ s.t. $1v=v$ for all $v\in V$

Subspaces

• A subspace is a subset of a vector space that forms a vector space under the same operations.
• A non-empty subset $W$ of a vector space $V$ is called a subset of $V$ if $W$ is a vector space under the same scalar field $F$ and the same operations of vector addition and scalar multiplication as in $V$.
• If $W$ is a non-empty subset of a vector space $V$, then $W$ is a subspace of $V$ iff $W$ satisfies closure under addition and closure under scalar multiplication.

Span

• Vector spaces and their subspaces share the crucial property of closure under arbitrary linear combination.
• Given a set $S=\{v_1,v_2,\dots ,v_n\}$ of vectors from vector space $V$, and let $W$ be the subspace of $V$ that contains all possible linear combinations of vectors in $S$, then $W$ is the span of $S$.

$W=\mathrm{Span}(S)=\mathrm{Span}\{v_1,v_2,\dots ,v_n\}$

or $S$ spans $W$.

Eg: $i=(1,0)$ and $j=(0,1)$ span ${\mathbb{R}}^2$.

Consider general vector $(a,b)\in {\mathbb{R}}^2$, $a,b\in \mathbb{R}$.

$(a,b)=a(1,0)+b(0,1)=ai+bj$

Since every vector in ${\mathbb{R}}^2$ can be written as a linear combination of $i$ and $j$, we conclude $\{i,j\}$ spans ${\mathbb{R}}^2$.