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Scientific claim

Ray in a Complex Vector Space Definition

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Claim

Let VV be a complex vector space and let ψ∈V\psi\in V be a non-zero vector.

The ray generated by ψ\psi is the set of all non-zero complex scalar multiples of ψ\psi:

[ψ]={cψ | c∈C∖{0}}.[\psi] = \left\{ c\psi \ \middle|\ c\in\mathbb{C}\setminus\{0\} \right\}.
Theoretical: Not applicableExperimental: Not applicable

Assumptions

No assumptions recorded.

Proof

This is a definition, so it is not proved from more fundamental propositions within this node.

Let VV be a vector space over the complex numbers C\mathbb{C}.

A vector ψ∈V\psi\in V is non-zero when:

ψ≠0.\psi\neq0.

For every complex number c∈Cc\in\mathbb{C}, scalar multiplication produces another vector cψ∈Vc\psi\in V.

The ray generated by ψ\psi contains every vector obtained by multiplying ψ\psi by a non-zero complex scalar:

[ψ]={cψ | c∈C∖{0}}.[\psi] = \left\{ c\psi \ \middle|\ c\in\mathbb{C}\setminus\{0\} \right\}.

The scalar c=0c=0 is excluded because:

0ψ=0.0\psi=0.

If the zero vector were included, every ray would contain the same zero vector. Excluding it ensures that distinct rays do not overlap.

Two non-zero vectors ϕ,ψ∈V\phi,\psi\in V belong to the same ray exactly when one is a non-zero complex scalar multiple of the other:

[ϕ]=[ψ]  ⟺  ∃c∈C∖{0} such that ϕ=cψ.[\phi]=[\psi] \iff \exists c\in\mathbb{C}\setminus\{0\} \text{ such that } \phi=c\psi.

A ray can also be understood as a one-dimensional complex subspace with its zero vector removed.

The associated one-dimensional subspace is:

span⁡{ψ}={cψ | c∈C}.\operatorname{span}\{\psi\} = \left\{ c\psi \ \middle|\ c\in\mathbb{C} \right\}.

The ray is therefore:

[ψ]=span⁡{ψ}∖{0}.[\psi] = \operatorname{span}\{\psi\}\setminus\{0\}.

Unlike a geometric half-line in real Euclidean space, a complex ray contains all non-zero complex multiples of ψ\psi. This includes changes in magnitude and multiplication by complex phases such as eiθe^{i\theta}.

Examples

  1. Ray Generated by a Vector in $\mathbb{C}^2$

    Let:

    V=C2V=\mathbb{C}^2

    and let:

    ψ=(1,i).\psi=(1,i).

    The ray generated by ψ\psi is:

    [ψ]={(c,ci) | c∈C∖{0}}.[\psi] = \left\{ (c,ci) \ \middle|\ c\in\mathbb{C}\setminus\{0\} \right\}.

    For example, the vectors (2,2i)(2,2i) and (−i,1)(-i,1) belong to [ψ][\psi] because:

    (2,2i)=2(1,i)(2,2i)=2(1,i)(−i,1)=−i(1,i).(-i,1)=-i(1,i).

  2. Ray Generated by a Basis Vector in $\mathbb{C}^3$

    Let:

    V=C3V=\mathbb{C}^3

    and let:

    e1=(1,0,0).e_1=(1,0,0).

    The ray generated by e1e_1 is:

    [e1]={(c,0,0) | c∈C∖{0}}.[e_1] = \left\{ (c,0,0) \ \middle|\ c\in\mathbb{C}\setminus\{0\} \right\}.

    Every non-zero vector in this ray has only its first component different from zero.

  3. Ray Generated by a Complex Function

    Let VV be a complex vector space of functions on R\mathbb{R}, and let:

    ψ(x)=eix.\psi(x)=e^{ix}.

    The ray generated by ψ\psi is the set of functions:

    [ψ]={f | f(x)=ceix, c∈C∖{0}}.[\psi] = \left\{ f \ \middle|\ f(x)=ce^{ix},\ c\in\mathbb{C}\setminus\{0\} \right\}.

    For example, both 2eix2e^{ix} and ieixie^{ix} belong to the same ray as eixe^{ix}.

Evidence

01
Is a ray in Hilbert space the same thing as a vector?joseph h · 2021-10-07 · supporting

Related Concepts

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