What is the Difference Between Vectors and Covectors?

[cathalcummins]

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.

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

<br /> V=\mathbb{R}^3<br />

V is the set

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

with basis, say

<br /> e_1=\left(<br /> \begin {array}{c}<br /> 1 \\<br /> \noalign{\medskip}<br /> 0 \\<br /> \noalign{\medskip}<br /> 0 \\<br /> \end {array}<br /> \right)<br />

<br /> e_2=\left(<br /> \begin {array}{c}<br /> 0 \\<br /> \noalign{\medskip}<br /> 1 \\<br /> \noalign{\medskip}<br /> 0 \\<br /> \end {array}<br /> \right)<br />


<br /> e_3=\left(<br /> \begin {array}{c}<br /> 0 \\<br /> \noalign{\medskip}<br /> 0 \\<br /> \noalign{\medskip}<br /> 1 \\<br /> \end {array}<br /> \right),<br />

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

<br /> v=(1,2,-5)^T<br />

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

<br /> v=1\cdot e_1+2\cdot e_2-5\cdot e_3<br />

Now define V^*, a space dual to V, by its elements f;
<br /> V^*= \{ f: f=f_i e^i=(x,y,z) \}<br />

so that f_1=x, f_2=y and f_3=z.

with (covariant) basis:


<br /> e_1=\left(<br /> \begin {array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> \end {array}<br /> \right)<br />

<br /> e_2=\left(<br /> \begin {array}{ccc}<br /> 0 &amp; 1 &amp; 0 \\<br /> \end {array}<br /> \right)<br />

<br /> e_3=\left(<br /> \begin {array}{ccc}<br /> 0 &amp; 0 &amp; 1 \\<br /> \end {array}<br /> \right)<br />


where it is demanded that

e^i(e_j)=\delta^i_j.


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

<br /> f(v)=f(v^i e_i)=v^i f(e_i)=v^i \delta^j_i f(e_j)<br />

But by our previous demand we have:

<br /> f(v)=v^i (e^j(e_i)) f(e_j)<br />

By linearity we have:

<br /> f(v)=v^i (e^j(e_i)) f(e_j)<br />

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

<br /> f(v)=f(e_j) v^i (e^j(e_i))=f(e_j) v^i (e^j(e_i))<br />

<br /> =(f(e_j) v^i e^j)(e_i)=(f(e_j) e^j)(v^i e_i)=(f(e_j) e^j)(v)<br />


As this is true \forall v \in V we must have:

}<br /> f \equiv f(e_j) e^j<br />

For notational purposes we define f_j=f(e_j). So that;
<br /> f \equiv f_j e^j<br />


So back to the problem at hand:

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

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

<br /> =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)<br />

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

=f_1 (1)+f_2 (2 )+f_3 (-5)

And so

f(v)=x (1)+y (2 )+z (-5)=x+2y-5z

So f(v) is a plane.


Right so my questions are:


1. What does f(v) being a plane mean?

2. I know that f(e_j)=f_j is just notation, and that it's form may be deduced from the given expression for f and the fact that the bases of V and V^* abide e^i(e_j)=\delta^i_j but what does f(e_j) mean? Is it just f acting on the basis elements of V? I mean, If i try to work out what f(e_k) is from f(v)=f(e_j)e^j(v)[/itex], we just get a cyclic definition f(e_k)=f(e_j)e^j(e_k)=f(e_j)\delta^k_j=f(e_k). And if so, I am finding it hard to define, say f(e_3)\equiv f_3=z. I mean would this be a valid description:<br /> <br /> f(e_i) is &quot;all of f&quot; acting on the i-th basis component of the corresponding vector space. It is defined by producing the i-th component of the covector f <br /> <br /> If anyone can clarify I&#039;d be ever so grateful.<br /> <br /> 3. The form of f I chose, relates to some sort of cartesian projection I think. Could someone shed some light on the situation.<br /> <br /> <br /> Cheers,<br /> <br /> <br /> <br /> edit: adjusted as requested.

 

[Chris Hillman]

As a courtesy to other PF readers, I'd encourage you to learn how to take advantage of the latex markup feature in VB, the software used in PF; see https://www.physicsforums.com/misc.php?do=bbcode . After reading a few lines from that page, you should have no trouble reformatting your document.

 

[Hurkyl]

If V is a vector space over R, then it's dual space V^* is, by definition, the space of all linear transformations from V to R. (also called linear functionals)

If we are studying a particular vector space, then it is conventional to call an element of its dual space a "covector".


If we choose a basis for V, then there is a natural choice of basis for V^*; we call it the dual basis. And, generally, we write the coordinates of a vector as a column array and the coordinates of a covector as a row array. The reason is as follows:

Let v \in V and \omega \in V^*.
Let [v] be the coordinate representation of v with respect to our chosen basis, and similarly for [\omega].

Then, we have:
[\omega(v)] = [\omega][v]

(The left hand side is the 1x1 matrix containing the number \omega(v). The right hand side is the product from matrix algebra, which yields a 1x1 matrix)

 

[cathalcummins]

>> If V is a vector space over R, then it's dual space V^* is, by definition, the space of all linear transformations from V to R. (also called linear functionals)

Okay

>> If we are studying a particular vector space, then it is conventional to call an element of its dual space a "covector".

Okay>> If we choose a basis for V, then there is a natural choice of basis for V^*; we call it the dual basis. And, generally, we write the coordinates of a vector as a column array and the coordinates of a covector as a row array.

Uh huh

>> The reason is as follows:

Let v \in V and \omega \in V^*.
Let [v] be the coordinate representation of v with respect to our chosen basis, and similarly for [\omega].

Then, we have:
[\omega(v)] = [\omega][v]

(The left hand side is the 1x1 matrix containing the number \omega(v). The right hand side is the product from matrix algebra, which yields a 1x1 matrix)

Okay

All that makes sense. It has given me some strength to know I am on the right track. However, my questions are still open, as far as my mind can stretch presently.

 

[clarkyeah]

For your first question, i give you an intutive sense of what 'the plane' means.
Assuming we are unknown of what shape the Earth has. But by some experiments, we derive a function f(v), which return the potential energy for an object, where v could be any point on earth. Then f(v)=0 is where all points with zero potential energy. As you obtain that f(v)=0 is a plane, we may say the Earth is flat.
The above explanation may not be strict. If i made any mistakes, please tell me.

 

[OrderOfThings]

Not just arrows sitting at an origin are examples of vectors. There are other objects that also form vector spaces. Take a set of equally spaced lines in the plane. Enumerate the lines.

Can you think of what it means to:

• 
  Multiply such a set of lines with a scalar? (easy)

• 
  Add two such sets of lines together? (slightly more complicated)

What does the resulting set of lines look like in each case?

Also, think a little about linear coordinate systems. Not in the way of basis vectors, but rather in the form of a parallelogramic grid. What does it mean to read off the coordinates of a vector from such a grid?