Okay, So I have am elementary question to ask but it is of fundamental importance to me. First things first, I have been looking through the posts on "the difference between vectors and covectors'' and found them to be helpful. But not too conducive to the way I am trying to learn about them. The posts seem to revolve around tangent and cotangent spaces. Although I will eventually go on to use my definition of covectors and vectors to define natural bases on manifolds, I am trying to ascertain a ``stand alone'' version of the defintion of covectors and vectors.(adsbygoogle = window.adsbygoogle || []).push({});

I have begun with good ol' reliable [itex]\mathbb{R}^3[/itex] for my vector space: Let us define a vector space [itex]V[/itex] such that:

[tex]

V=\mathbb{R}^3

[/tex]

[tex]V[/tex] is the set

[tex]

V=\{ v:v^i e_i=(v^1,v^2,v^3)^T | v^i \in \mathbb{R}^3 \}

[/tex]

with basis, say

[tex]

e_1=\left(

\begin {array}{c}

1 \\

\noalign{\medskip}

0 \\

\noalign{\medskip}

0 \\

\end {array}

\right)

[/tex]

[tex]

e_2=\left(

\begin {array}{c}

0 \\

\noalign{\medskip}

1 \\

\noalign{\medskip}

0 \\

\end {array}

\right)

[/tex]

[tex]

e_3=\left(

\begin {array}{c}

0 \\

\noalign{\medskip}

0 \\

\noalign{\medskip}

1 \\

\end {array}

\right),

[/tex]

To be more explicit, let me define what the vector is, say

[tex]

v=(1,2,-5)^T

[/tex]

So that [itex]v^1=1[/itex], [itex]v^2=2[/itex] and [itex]v^3=-5[/itex]. And so that:

[tex]

v=1\cdot e_1+2\cdot e_2-5\cdot e_3

[/tex]

Now define [itex]V^*[/itex], a space dual to [itex]V[/itex], by its elements [itex]f[/itex];

[tex]

V^*= \{ f: f=f_i e^i=(x,y,z) \}

[/tex]

so that [itex]f_1=x[/itex], [itex]f_2=y[/itex] and [itex]f_3=z[/itex].

with (covariant) basis:

[tex]

e_1=\left(

\begin {array}{ccc}

1 & 0 & 0 \\

\end {array}

\right)

[/tex]

[tex]

e_2=\left(

\begin {array}{ccc}

0 & 1 & 0 \\

\end {array}

\right)

[/tex]

[tex]

e_3=\left(

\begin {array}{ccc}

0 & 0 & 1 \\

\end {array}

\right)

[/tex]

where it is demanded that

[tex]e^i(e_j)=\delta^i_j[/tex].

Further, if we want to know how the [itex]f \in V^*[/itex] acts on the [itex]v \in V[/itex], we must derive a relation:

[tex]

f(v)=f(v^i e_i)=v^i f(e_i)=v^i \delta^j_i f(e_j)

[/tex]

But by our previous demand we have:

[tex]

f(v)=v^i (e^j(e_i)) f(e_j)

[/tex]

By linearity we have:

[tex]

f(v)=v^i (e^j(e_i)) f(e_j)

[/tex]

Now [itex]v^i,f(e_j) \in \mathbb{R}[/itex] so we can just shift them around at will.

[tex]

f(v)=f(e_j) v^i (e^j(e_i))=f(e_j) v^i (e^j(e_i))

[/tex]

[tex]

=(f(e_j) v^i e^j)(e_i)=(f(e_j) e^j)(v^i e_i)=(f(e_j) e^j)(v)

[/tex]

As this is true [itex]\forall v \in V[/itex] we must have:

[tex]}

f \equiv f(e_j) e^j

[/tex]

For notational purposes we define [itex]f_j=f(e_j)[/itex]. So that;

[tex]

f \equiv f_j e^j

[/tex]

So back to the problem at hand:

[tex]

f(v) = f_je^j(v)=f_1 e^1(v)+f_2 e^2(v)+f_3 e^3(v)

[/tex]

[tex]

f(v) = f_je^j(1\cdot e_1+2\cdot e_2-5\cdot e_3)

[/tex]

[tex]

=f_1 e^1(1\cdot e_1+2\cdot e_2-5\cdot e_3)+f_2 e^2(1\cdot e_1+2\cdot e_2-5\cdot e_3)+f_3 e^3(1\cdot e_1+2\cdot e_2-5\cdot e_3)

[/tex]

[tex]=f_1 e^1(1\cdot e_1)+f_2 e^2(2\cdot e_2)+f_3 e^3(-5\cdot e_3)[/tex]

[itex]=f_1 (1)+f_2 (2 )+f_3 (-5)[/itex]

And so

[itex]f(v)=x (1)+y (2 )+z (-5)=x+2y-5z[/itex]

So [tex]f(v)[/tex] is a plane.

Right so my questions are:

1. What does [itex]f(v)[/itex] being a planemean?

2. I know that [itex]f(e_j)=f_j [/itex] is just notation, and that it's form may bededucedfrom the given expression for [itex]f[/itex] and the fact that the bases of [itex]V[/itex] and [itex]V^*[/itex] abide [itex]e^i(e_j)=\delta^i_j[/itex] but what does [itex]f(e_j)[/itex]mean? Is it just [itex]f[/itex] acting on the basis elements of [itex]V[/itex]? I mean, If i try to work out what [itex]f(e_k)[/itex] is from [tex]f(v)=f(e_j)e^j(v)[/itex], we just get a cyclic definition [itex]f(e_k)=f(e_j)e^j(e_k)=f(e_j)\delta^k_j=f(e_k)[/itex]. And if so, I am finding it hard to define, say [itex]f(e_3)\equiv f_3=z[/itex]. I mean would this be a valid description:

[itex]f(e_i)[/itex] is "all of [itex]f[/itex]" acting on the i-th basis component of the corresponding vector space. It is defined by producing the i-th component of the covector [itex]f[/itex]

If anyone can clarify I'd be ever so grateful.

3. The form of [itex]f[/itex] I chose, relates to some sort of cartesian projection I think. Could someone shed some light on the situation.

Cheers,

edit: adjusted as requested.

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# Vectors and Co-vectors

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