Vector space

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

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

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

Definitions

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

Theorems

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