Difference between revisions of "Dual vector space"
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(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) | (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 | + | 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 | * <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> | *: 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"/> | * <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"/> | ||
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==Dual basis== | ==Dual basis== | ||
Revision as of 17:21, 5 June 2015
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
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
