A vector space is an algebraic structure consisting of an additive Abelian group ${\displaystyle V}$ (elements of which are called vectors, and are denoted in bold), a field ${\displaystyle F}$ (elements of which are called scalars), and a scalar multiplication function ${\displaystyle \times:F\times V\to V}$ following these properties:

• Distributive property of scalar multiplication over vector addition: For all ${\displaystyle a \in F}$ and ${\displaystyle \mathbf v, \mathbf w \in V}$ , ${\displaystyle a(\mathbf v+\mathbf w)=a\mathbf v+a\mathbf w}$ .
• Distributive property of scalar multiplication over field addition: For all ${\displaystyle a, b \in F}$ and ${\displaystyle \mathbf v \in V}$ , ${\displaystyle (a+b)\mathbf v=a\mathbf v+b\mathbf v}$ .
• Associative law of combined scalar and field multiplication: For all ${\displaystyle a, b \in F}$ and ${\displaystyle \mathbf v \in V}$ , ${\displaystyle a(b\mathbf v)=(ab)\mathbf v}$ .
• Scalar multiplication identity: With 1 as the field multiplicative identity, for all ${\displaystyle \mathbf v \in V}$ , we have ${\displaystyle 1\mathbf v = \mathbf v}$ .

As both ${\displaystyle V}$ and ${\displaystyle F}$ each have their own respective additive identites, we will denote boldface ${\displaystyle \mathbf{0}}$ to represent the additive identity in ${\displaystyle V}$ . We then say that ${\displaystyle V}$ is a vector space over the field ${\displaystyle F}$ .

Definitions

• A subset ${\displaystyle S\subseteq V}$ is:
  • Linearly dependent if there exist (distinct) vectors ${\displaystyle \mathbf v_1,\ldots,\mathbf v_n\in S}$ and scalars ${\displaystyle a_1,\ldots,a_n\in F}$ , with at least one of these scalars non-zero, such that ${\displaystyle \sum_{k=1}^n a_k\mathbf v_k=a_1\mathbf v_1+\cdots+a_n\mathbf v_n=\mathbf 0}$ . Otherwise, we say ${\displaystyle S}$ is linearly independent.
  • A spanning set if, for any ${\displaystyle \mathbf v \in V}$ there exist vectors ${\displaystyle \mathbf v_1,\ldots,\mathbf v_n\in S}$ and scalars ${\displaystyle a_1,\ldots,a_n\in F}$ such that ${\displaystyle \sum_{k=1}^n a_k\mathbf v_k=a_1\mathbf v_1+\cdots+a_n\mathbf v_n=\mathbf v}$ . That is to say, any vector in ${\displaystyle V}$ is a linear combination of vectors in ${\displaystyle S}$ .
  • A basis for ${\displaystyle V}$ if ${\displaystyle S}$ is linearly independent and a spanning set.
  • A subspace for ${\displaystyle V}$ if ${\displaystyle S}$ is also a subgroup and is closed under scalar multiplication: For any vector ${\displaystyle \mathbf w\in S}$ and scalar ${\displaystyle a \in F}$ , ${\displaystyle a\mathbf w\in S}$ .
• Given two vector spaces ${\displaystyle V}$ and ${\displaystyle W}$ over a field ${\displaystyle F}$ , a function ${\displaystyle T:V\to W}$ is a linear transformation if for any ${\displaystyle \mathbf u,\mathbf v,\in V}$ and ${\displaystyle a \in F}$ , ${\displaystyle T(\mathbf u+\mathbf v)=T(\mathbf u)+T(\mathbf v)}$ and ${\displaystyle T(a\mathbf u)=aT(\mathbf u)}$ . In abstract algebra, this is known as a homomorphism.

Theorems

• If ${\displaystyle \mathcal{B}}$ and ${\displaystyle \mathcal{C}}$ are bases for ${\displaystyle V}$ , then they have the same cardinality. That is, there exists a bijective function ${\displaystyle f:\mathcal B\to\mathcal C}$ (proof). As such, we may define the dimension of ${\displaystyle V}$ as the cardinality of any basis for ${\displaystyle V}$ .
• Given a basis ${\displaystyle \mathcal B=\{\mathbf v_1,\mathbf v_2,\mathbf v_3,\ldots\}}$ , and linear combinations ${\displaystyle \sum_{k=1}^n a_k\mathbf v_k}$ and ${\displaystyle \sum_{k=1}^n b_k\mathbf v_k}$ , if ${\displaystyle \sum_{k=1}^n a_k\mathbf v_k=\sum_{k=1}^n b_k\mathbf v_k}$ , then ${\displaystyle a_k=b_k}$ for each applicable ${\displaystyle k}$ . This is to say any vector in ${\displaystyle V}$ is a unique linear combination of vectors in ${\displaystyle \mathcal{B}}$ .