Components of a dual vector?

  • #1

Main Question or Discussion Point

Hi everyone,

So I'm going through a chapter on dual spaces and I came across this:

"A key property of any dual vector ##f## is that it is entirely determined by its values on basis vectors.

## f_i \equiv f(e_i) ##

which we refer to as the components of ##f## in the basis ##{e_i}##, this is justified by

##e^i(e_j) = \delta^i_j## "

-- This doesn't make any sense to me. How are the components of the dual space only dependent on the basis of V? Wouldn't that mean the components are always the same? I thought the whole point of components was that they varied.

Also, is the second equation assuming a cartesian basis? (not sure what the technical term is) Because I can think of some orthonormal bases for which this doesn't hold. i.e. I can think of some orthonormal basis where 'picking off' the ith component does not yield 1.

I'm obviously confused.

Dual spaces in general just confuse me, I understand that the dual space is the set of linear functionals on V, and that they can be represented as 1-forms, but as far as the details go (coordinates and bases of the dual space) I'm completely lost. Any help would be much appreciated.
 

Answers and Replies

  • #2
RUber
Homework Helper
1,687
344
Using the Riesz Representation of functionals in an inner-product space, you can express any functional as an inner product with a vector in the space. Implying that you would use the same basis set to express the representation (i.e. vector form).
https://en.wikipedia.org/wiki/Riesz_representation_theorem
 
  • #3
Using the Riesz Representation of functionals in an inner-product space, you can express any functional as an inner product with a vector in the space. Implying that you would use the same basis set to express the representation (i.e. vector form).
https://en.wikipedia.org/wiki/Riesz_representation_theorem
Could you expand what you mean by 'vector form'? Are you talking about a differential form (covector)?
 
  • #4
RUber
Homework Helper
1,687
344
The vector form I was referring to is just the ##f_i = f(e_i)## notation you used.
 
  • Like
Likes mattclgn

Related Threads on Components of a dual vector?

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
1
Views
616
Replies
12
Views
4K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
6
Views
26K
  • Last Post
Replies
19
Views
6K
Replies
4
Views
634
Replies
1
Views
1K
Top