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Vector Spaces

Abstract

The conditions for a vector space.

Keywords:VectorsVector Spaces

A vector space, defined over a field F\mathbb{F}, is a set VV whose elements can be added together and multiplied by a scalars (F1\mathbb{F}^{1}). The elements of VV, whilst often called vectors, can be a variety of different mathematical objects, such as vectors in the typical sense, or matrices.

Vector spaces defined over R\mathbb{R} are known as real vector spaces, whilst vector spaces defined over C\mathbb{C} are defined as complex vector spaces.

Vector Space Axioms

Let ψ, σ, ω ∈ V\psi, ~\sigma, ~\omega~\in~V and a,b∈F1a,b \in \mathbb{F}^{1}.

In order for VV to be a vector space, the addition and scalar multiplication operations must meet a set of Axioms, defined below.

The addition operation must obey the following conditions

  1.   ψ+σ=σ+ψ~~\psi + \sigma = \sigma + \psi
    • Commutativity
  2. ψ+(σ+ω)=(ψ+σ)+ω\psi + (\sigma + \omega) = (\psi + \sigma) + \omega
    • Associativity
  3. ∃ I∈ V\exists ~I \in ~V such that ψ+I=ψ\psi + I = \psi.
    • Identity
  4.   ∀ v ∈ V,∃−v ∈ V~~\forall~v~\in~V, \exists -v~\in~V such that v+−v=Iv+-v=I.
    • Inverse

The multiplication by a scalar operation must obey the following conditions

  1. a(ψ+ω)=aψ+bωa(\psi + \omega) = a \psi + b \omega
    • distributive over VV
  2. (a+b)ψ=aψ+bψ(a+b)\psi = a\psi + b\psi
    • distributive over scalar

Further Vector Space Properties

Linear Combination
Linear (In)dependence
Linear Subspace
Basis

Let a1, a2,… an ∈ V ∀ na_{1}, ~a_{2}, \ldots ~a_{n}~\in~V~\forall~n and λ1, λ2,… λn ∈ F1 ∀ n\lambda_1,~\lambda_2, \ldots ~\lambda_n~\in~\mathbb{F}^{1} ~\forall~n

A linear combination of the vectors {a1, a2,…, an}\{ a_{1}, ~a_{2}, \ldots, ~a_{n} \} is given by

a~=λ1a1+λ2a2…λnan\tilde{a} = \lambda_1 a_{1} + \lambda_2 a_2 \ldots \lambda_n a_n
(1)

All linear combinations of vectors in VV are also in V, a~ ∈ V\tilde{a}~\in~V.

Subspaces

Let VV be a vector space over F\mathbb{F}. A subset U⊂VU \subset V is a subspace of VV if the following conditions hold

  1. U≠∅U \neq \emptyset
    • The subset is not empty.
  2. u+v ∈ U ∀ u,v ∈ Uu+v~\in~U~\forall~u,v~\in~U.
    • UU is closed under addition.
  3. λu ∈ U ∀ u ∈ U,λ ∈ F1\lambda u~\in~U~\forall~u~\in~U, \lambda~\in~\mathbb{F}^{1}
    • UU is closed under multiplication by scalars.

Properties and Operations of Subspaces

Union of Subspaces
Direct Sum

Let VV be a vector space and UU and WW be subsets of VV, U,W ⊂ VU,W~\subset~V.

The union of UU and WW is also a subspace of VV:

U∩W⊂VU \cap W \subset V
(5)
Linear Algebra
Matrices
Linear Algebra
Basis