Difference between revisions of "Dual vector space"
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| − | + | ==Definition== | |
| + | Given a [[Vector space|vector space]] {{M|(V,F)}} we define the '''dual''' or '''conjugate''' vector space<ref name="LAVEP">Linear Algebra via Exterior Products - Sergei Winitzki</ref> (which we denote {{M|V^*}}) as: | ||
| + | * <math>V^*=\text{Hom}(V,F)</math> (recall this the set of all [[Homomorphism|homomorphisms]] (specifically [[Linear map|linear ones]]) from {{M|V}} to {{M|F}}) | ||
| + | * That is <math>V^*=\{f:V\rightarrow F|\ f\text{ is linear}\}</math> (which is to say <math>f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\ \forall x,y\in V\ \forall \alpha,\beta\in F</math>) | ||
| + | We usually denote the elements of {{M|V^*}} with {{M|*}}s after them, that is something like <math>f^*\in V^*</math> | ||
| − | + | We call the elements of {{M|V^*}}: | |
| + | * [[Covector|Covectors]] | ||
| + | * Dual vectors | ||
| + | * Linear form | ||
| + | * Linear functional (and {{M|V^*}} the set of ''linear functionals'')<ref>Introduction to Smooth Manifolds - John M Lee - I THINK! CHECK THIS - CERTAINLY have seen it somewhere though</ref> | ||
| + | ''Covectors'' and ''Dual vectors'' are interchangeable and I find myself using both without a second thought. | ||
| + | |||
| + | ==Equality of covectors== | ||
| + | We say two covectors, <math>f^*,g^*:V\rightarrow F</math> are equal if<ref name="LAVEP"/>: | ||
| + | * <math>\forall v\in V[f^*(v)=g^*(v)]</math><ref group="note">I avoid the short form: <math>f^*v=f^*(v)</math> because the <math>*</math> looks too much like an operator</ref> - that is they agree on their domain (as is usual for function equality) | ||
| + | |||
| + | ==The zero covector== | ||
| + | The zero covector<math>:V\rightarrow F</math> with <math>v\mapsto 0</math><ref name="LAVEP"/> | ||
| + | {{Todo|Confirm it is written {{M|0^*}} before writing that here | ||
| + | |||
| + | ==Examples== | ||
| + | ===Dual vectors of {{M|\mathbb{R}^2}}=== | ||
| + | Consider (for {{M|v\in\mathbb{R}^2}}: <math>f^*(v)=2x</math> and <math>g^*(v)=y-x</math> - it is easy to see these are linear and thus are covectors!<ref group="note" name="RIP">Example shamelessly ripped</ref><ref name="LAVEP"/> | ||
| + | ===Dual vectors of {{M|\mathbb{R}[x]_{\le 2} }}=== | ||
| + | (Recall that <math>\mathbb{R}[x]_{\le 2}</math> denotes all polynomials up to order 2, that is: <math>\alpha+\beta x+\gamma x^2</math> for <math>\alpha,\beta,\gamma\in\mathbb{R}</math> in this case) | ||
| + | |||
| + | Consider {{M|p\in\mathbb{R}[x]_{\le 2} }} and <math>f*,g^*:\mathbb{R}[x]_{\le 2}\rightarrow\mathbb{R}</math> given by: | ||
| + | * <math>f^*(p)=\int^{+\infty}_0e^{-x}p(x)dx</math> - this is a covector | ||
| + | *: To see this notice: <math>f^*(\alpha p+\beta q)=\int^{+\infty}_0e^{-x}[\alpha p(x)+\beta q(x)]dx</math><math>=\alpha\int^{+\infty}_0e^{-x}p(x)dx+\beta\int^{+\infty}_0e^{-x}q(x)dx</math> | ||
| + | * <math>g^*(p)=\frac{dp}{dx}\Big|_{x=1}</math> (note that: <math>g^*(\alpha+\beta x+\gamma x^2)=\beta+2\gamma</math> - so this isn't just projecting to coefficients) is also a covector<ref name="RIP" group="note"/><ref name="LAVEP"/> | ||
| + | |||
| + | ==Dual basis== | ||
| + | {{Begin Theorem}} | ||
| + | Theorem: Given a basis {{M|1=\{e_1,\cdots,e_n\} }} of a vector space {{M|(V,F)}} there is a corresponding basis to {{M|V^*}}, {{M|1=\{e_1^*,\cdots,e_n^*\} }} where each {{M|e^*_i}} is the {{M|i^\text{th} }} coordinate of a vector {{M|v\in V}} (that is the coefficient of {{M|e_i}} when {{M|v}} is expressed as {{M|1=\sum^n_{j=1}v_je_j}}). That is to say that {{M|e_i^*}} [[Projector|projects]] {{M|v}} onto it's {{M|e_i^\text{th} }} coordinate. | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|Proof}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Covector]] | ||
| + | * [[Covector applied to a tensor product]] | ||
| + | |||
| + | ==Notes== | ||
| + | <references group="note"/> | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
| − | |||
Latest revision as of 05:35, 8 December 2016
Contents
Definition
Given a vector space [ilmath](V,F)[/ilmath] we define the dual or conjugate vector space[1] (which we denote [ilmath]V^*[/ilmath]) as:
- [math]V^*=\text{Hom}(V,F)[/math] (recall this the set of all homomorphisms (specifically linear ones) from [ilmath]V[/ilmath] to [ilmath]F[/ilmath])
- That is [math]V^*=\{f:V\rightarrow F|\ f\text{ is linear}\}[/math] (which is to say [math]f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\ \forall x,y\in V\ \forall \alpha,\beta\in F[/math])
We usually denote the elements of [ilmath]V^*[/ilmath] with [ilmath]*[/ilmath]s after them, that is something like [math]f^*\in V^*[/math]
We call the elements of [ilmath]V^*[/ilmath]:
- Covectors
- Dual vectors
- Linear form
- Linear functional (and [ilmath]V^*[/ilmath] the set of linear functionals)[2]
Covectors and Dual vectors are interchangeable and I find myself using both without a second thought.
Equality of covectors
We say two covectors, [math]f^*,g^*:V\rightarrow F[/math] are equal if[1]:
- [math]\forall v\in V[f^*(v)=g^*(v)][/math][note 1] - that is they agree on their domain (as is usual for function equality)
The zero covector
The zero covector[math]:V\rightarrow F[/math] with [math]v\mapsto 0[/math][1] {{Todo|Confirm it is written [ilmath]0^*[/ilmath] before writing that here
Examples
Dual vectors of [ilmath]\mathbb{R}^2[/ilmath]
Consider (for [ilmath]v\in\mathbb{R}^2[/ilmath]: [math]f^*(v)=2x[/math] and [math]g^*(v)=y-x[/math] - it is easy to see these are linear and thus are covectors![note 2][1]
Dual vectors of [ilmath]\mathbb{R}[x]_{\le 2} [/ilmath]
(Recall that [math]\mathbb{R}[x]_{\le 2}[/math] denotes all polynomials up to order 2, that is: [math]\alpha+\beta x+\gamma x^2[/math] for [math]\alpha,\beta,\gamma\in\mathbb{R}[/math] in this case)
Consider [ilmath]p\in\mathbb{R}[x]_{\le 2} [/ilmath] and [math]f*,g^*:\mathbb{R}[x]_{\le 2}\rightarrow\mathbb{R}[/math] given by:
- [math]f^*(p)=\int^{+\infty}_0e^{-x}p(x)dx[/math] - this is a covector
- To see this notice: [math]f^*(\alpha p+\beta q)=\int^{+\infty}_0e^{-x}[\alpha p(x)+\beta q(x)]dx[/math][math]=\alpha\int^{+\infty}_0e^{-x}p(x)dx+\beta\int^{+\infty}_0e^{-x}q(x)dx[/math]
- [math]g^*(p)=\frac{dp}{dx}\Big|_{x=1}[/math] (note that: [math]g^*(\alpha+\beta x+\gamma x^2)=\beta+2\gamma[/math] - so this isn't just projecting to coefficients) is also a covector[note 2][1]
Dual basis
Theorem: Given a basis [ilmath]\{e_1,\cdots,e_n\}[/ilmath] of a vector space [ilmath](V,F)[/ilmath] there is a corresponding basis to [ilmath]V^*[/ilmath], [ilmath]\{e_1^*,\cdots,e_n^*\}[/ilmath] where each [ilmath]e^*_i[/ilmath] is the [ilmath]i^\text{th} [/ilmath] coordinate of a vector [ilmath]v\in V[/ilmath] (that is the coefficient of [ilmath]e_i[/ilmath] when [ilmath]v[/ilmath] is expressed as [ilmath]\sum^n_{j=1}v_je_j[/ilmath]). That is to say that [ilmath]e_i^*[/ilmath] projects [ilmath]v[/ilmath] onto it's [ilmath]e_i^\text{th} [/ilmath] coordinate.
TODO: Proof
See also
Notes
- ↑ I avoid the short form: [math]f^*v=f^*(v)[/math] because the [math]*[/math] looks too much like an operator
- ↑ 2.0 2.1 Example shamelessly ripped
