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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 \rightarrow 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 \left(\mathbf v + \mathbf w \right) = 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, \left(a + b\right) \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 \left(b \mathbf v\right) = \left(ab\right)\mathbf v.
  • Scalar multiplication identity: With 1 as the field multiplicative identity, for all v ∈ 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.

DefinitionsEdit

  • A subset S \subseteq V is:
    • Linearly dependent if there exist (distinct) vectors \mathbf v_1,\mathbf v_2,\dots,\mathbf v_n \in S and scalars a_1,a_2,\dots,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 + a_2\mathbf v_2+\dots+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,\mathbf v_2,\dots,\mathbf v_n \in S and scalars a_1,a_2,\dots,a_n \in F such that \sum_{k=1}^n a_k\mathbf v_k = a_1\mathbf v_1 + a_2\mathbf v_2+\dots+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 \rightarrow W is a linear transformation if for any \mathbf u,\mathbf v, \in V and a \in F, T\left( \mathbf u + \mathbf v \right) = T\left( \mathbf u \right) + T\left(\mathbf v \right) and T\left(a \mathbf u \right) = aT\left( \mathbf u \right). In abstract algebra, this is known as a homomorphism.

TheoremsEdit

  • 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 \rightarrow \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 = \left\{\mathbf v_1, \mathbf v_2, \mathbf v_3, \dots \right\}, 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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