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Vector space

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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 .

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