FANDOM


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 $ \mathbf0 $ 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 $ .