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Thread: Covectors

  1. #1 Covectors 
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    Can anyone give me a little help understanding the fundamentals of covectors vis a vis (contravariant) vectors ,vector spaces, mapping to the Real numbers etc.?

    If there is I will give a specific question or two where I am struggling.


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  3. #2  
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    https://en.wikipedia.org/wiki/Linear_form

    try this


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  4. #3  
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    Quote Originally Posted by mathman View Post
    I have already looked at that as well as others.

    What I want to clear up in my head at the moment is , when we picture vectors and covectors in a geometrical way what might be an example of a 3 dimensional covector acting on ,say a 3 dimensional vector (3,4,5 for example)?

    Would any covectors such as -3,-4 ,0 work since the covector added to the vector gives a real number (ie 5)?

    Could it also be -3,-4 ,n which would give the real number n+5 ?

    I think I may have understood that it is better to study this subject mathematically rather than geometrically but I am trying to find my footing.

    By the way I am also dipping into Markus Hanke's web site as it seems very good.
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  6. #5  
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    @Giant Evil That is nice, but running through the episodes it doesn't appear to me that he mentions covectors.

    Was my understanding in post#3 correct?

    I always feel the need to produce a specific example when I am learning a subject in generalities ,in order to reassure myself that my initial steps in understanding are based in a reality.

    I also have the vice of turning the pages to the middle of the book which might explain why I am trying to understand (well,hopefully revise) the basics of vectors and matrices whilst also trying to learn about tensors and General Relativity!
    Last edited by geordief; March 2nd, 2019 at 07:13 AM.
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    I have (also) done the first 5 videos of that eigenchris series.

    It is a slog and that is why I posted my question in post#3- to reassure myself that my understanding of what a covector actually is or does.

    Before I continue on to the next lessons..
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  9. #8  
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    Well, I hadn’t even considered covectors(or whatever of the half dozen synonyms for such) untill I had read your post. Starting with the wiki article referenced, there are pages and pages of background information referenced for that article. So I’m still working on what specifically a covector is. I have some general ideas, but they’re incomplete. The basis for a general understanding of covectors definitely involves at least the rudiments of abstract algebra, analysis, and linear algebra. You will also need to understand the basic idea of a set as used in mathematics, first order logic, and axiomatic systems, to engage any of the base materials from abstract algebra, analysis, and linear algebra. Any advanced understanding of covectors is going to require topology. Good luck!
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    Maybe it is too simple, but I have the impression there are basically two kinds of vectors, covariant and contra-variant. The distinction is important in differential geometry.
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    I think the ordinary vectors are the ones we are all familiar with -that form parallelograms with the "destination" at the tip.

    The covectors seem to lie in the same direction but produced until a perpendicular raised from it goes through the same point as the one at the apex of the aforementioned parallelogram.

    It seems that you can combine these 2 kinds of vectors as a replacement for the Pythagoras theorem when there is no right angle to work with.

    But it also seems that these two kinds of vectors can be combined to form tensors at any point in the coordinate system.

    Tensors it seems are the same no matter what frame of reference is chosen and that is apparently why they are so useful in General Relativity.

    If I am right...
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    geodief, you need to start from the beginning (always the best place!).

    Suppose a vector space . Then there always exists another vector space, say , such that . So for ANY and ALL then .

    Elements in the space are variously called dual vectors, covectors or (god help us ) covariant vectors.

    As an irrelevant aside, the tensor product of 2 dual spaces is the bilinear mapping , so that for and that

    The most famous of these tensor products(in Relativity theory) is the so-called metric tensor
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    Quote Originally Posted by Guitarist View Post
    geodief, you need to start from the beginning (always the best place!).

    Suppose a vector space . Then there always exists another vector space, say , such that . So for ANY and ALL then .

    Elements in the space are variously called dual vectors, covectors or (god help us ) covariant vectors.

    As an irrelevant aside, the tensor product of 2 dual spaces is the bilinear mapping , so that for and that

    The most famous of these tensor products(in Relativity theory) is the so-called metric tensor
    Are all these vector spaces always infinite and continuous?

    Are there always vectors "between" any 2 vectors?
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    Are all these vector spaces always infinite and continuous?
    By no means, no. I remind you that the dimension of a vector space is simply the minimum number of basis vectors needed to define any other vector using, say, , where the are numbers, and the are the basis vectors.

    In general, infinite-dimensional vector spaces are complex-valued i.e. in the above the are complex numbers.

    Here's a puzzle for you: given a vector space of complex-valued continuous functions, what are the duals (or covectors)?
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  16. #15  
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    Quote Originally Posted by Guitarist View Post
    Are all these vector spaces always infinite and continuous?
    By no means, no. I remind you that the dimension of a vector space is simply the minimum number of basis vectors needed to define any other vector using, say, , where the are numbers, and the are the basis vectors.

    In general, infinite-dimensional vector spaces are complex-valued i.e. in the above the are complex numbers.

    Here's a puzzle for you: given a vector space of complex-valued continuous functions, what are the duals (or covectors)?
    Actually I was asking whether there could be an infinite number of elements in a typical vector space.

    I wasn't talking about the number of dimensions.

    I haven't learned much about complex vector spaces yet and doubt that I could answer your question - I suspect that even to understand the answer might be very hard going at this stage.

    By the way ,can I work with two dimensional vector spaces on a purely pedagogic level? (to understand more clearly what the dual space might be in that case)
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    Quote Originally Posted by geordief View Post

    Actually I was asking whether there could be an infinite number of elements in a typical vector space.
    Yes, in fact there almost always are. I'll explain in a minute.

    By the way ,can I work with two dimensional vector spaces on a purely pedagogic level? (to understand more clearly what the dual space might be in that case)
    Yes you can for the finite-dimensional case. Even experienced mathematicians do sometimes.

    So to your earlier question, using 2-dimensional vector space, suppose we write the vector where as before the are basis vectors and the just positive integers
    Suppose another vector in the same space, say . Then if the then .

    But since the are just numbers, the question becomes "how many positive integers are there".

    No sweeties for a correct answer!
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  18. #17  
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by geordief View Post

    Actually I was asking whether there could be an infinite number of elements in a typical vector space.
    Yes, in fact there almost always are. I'll explain in a minute.

    By the way ,can I work with two dimensional vector spaces on a purely pedagogic level? (to understand more clearly what the dual space might be in that case)
    Yes you can for the finite-dimensional case. Even experienced mathematicians do sometimes.



    So to your earlier question, using 2-dimensional vector space, suppose we write the vector where as before the are basis vectors and the just positive integers
    Suppose another vector in the same space, say . Then if the then .

    But since the are just numbers, the question becomes "how many positive integers are there".

    No sweeties for a correct answer!
    Can I (again for pedagogic reasons) take a finite subset of the infinite number of elements in a two dimensional vector space ?

    Let's say (1x +2y) , (1x + 1y) and (2x +1y).... ,using x and y as shorthand for bases as I don't have the notation available on my keyboard.

    Would its dual vector space also have 3 elements ? How many relationships could be formed between the elements of either set with the elements in the other?

    Perhaps I have misconstrued the whole approach now?
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  19. #18  
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    Quote Originally Posted by geordief View Post

    Can I (again for pedagogic reasons) take a finite subset of the infinite number of elements in a two dimensional vector space ?
    Yes you can - recall that for any vector space, the basis vectors are a subspace of that space - but why would you bother? I cannot see where it gets you

    Let's say (1x +2y) , (1x + 1y) and (2x +1y).... ,using x and y as shorthand for bases as I don't have the notation available on my keyboard.

    Would its dual vector space also have 3 elements ?
    I don't quite get it.
    A vector space and its dual space have the same dimension, that is the same number of basis vectors. The total number of vectors is immaterial unless you similarly restrict the field over which is defined.

    I'm afraid you are barking up the wrong gum tree
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  20. #19  
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by geordief View Post

    Can I (again for pedagogic reasons) take a finite subset of the infinite number of elements in a two dimensional vector space ?
    Yes you can - recall that for any vector space, the basis vectors are a subspace of that space - but why would you bother? I cannot see where it gets you

    Let's say (1x +2y) , (1x + 1y) and (2x +1y).... ,using x and y as shorthand for bases as I don't have the notation available on my keyboard.

    Would its dual vector space also have 3 elements ?
    I don't quite get it.
    A vector space and its dual space have the same dimension, that is the same number of basis vectors. The total number of vectors is immaterial unless you similarly restrict the field over which is defined.

    I'm afraid you are barking up the wrong gum tree
    So there is no one to one correspondence between the elements of a vector space and the elements of its dual space?

    They can't be paired up as it were ?(double helix fashion)
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    Quote Originally Posted by geordief View Post
    So there is no one to one correspondence between the elements of a vector space and the elements of its dual space?

    They can't be paired up as it were ?(double helix fashion)
    I told you 1 post ago that a vector space and its dual have the same dimension - the exact same number of basis vectors. They are therefore isomorphic.

    I suggest you try to construct a proof of this for yourself, rather than expecting to be endlessly spoon-fed.

    Good luck
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  22. #21  
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    although I am mathematician I do not anymore deal with pure mathematics. but I can recommend you some references though in pure math.

    1) Schaum's Outlines, Linear Algebra (highly advised)

    2) general algebra (see:bourbaki)

    3) introductory functional analysis and its applications (kreyszig)
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  23. #22  
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    Quote Originally Posted by geordief View Post
    Can anyone give me a little help understanding the fundamentals of covectors vis a vis (contravariant) vectors ,vector spaces, mapping to the Real numbers etc.?

    If there is I will give a specific question or two where I am struggling.
    A co-vector is defined as a linear map which maps a vector to be a scalar , that is . . .
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  24. #23  
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    Quote Originally Posted by trfrm View Post
    Quote Originally Posted by geordief View Post
    Can anyone give me a little help understanding the fundamentals of covectors vis a vis (contravariant) vectors ,vector spaces, mapping to the Real numbers etc.?

    If there is I will give a specific question or two where I am struggling.
    A co-vector is defined as a linear map which maps a vector to be a scalar , that is . . .
    If the vector lies in the [x,y] plane is the scalar that the covectors map it to the same as a vector with a zero value for either x or y?

    ie either [0,y] or [x,0]?

    Are there an infinite number of covectors for any particular vector we choose?
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  25. #24  
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    Quote Originally Posted by geordief View Post
    Are there an infinite number of covectors for any particular vector we choose?
    A vector corresponds only one co-vector , such that . . . A vector space and its dual vector space have same cardinality, that is . . .
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  26. #25  
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    Quote Originally Posted by trfrm View Post
    Quote Originally Posted by geordief View Post
    Are there an infinite number of covectors for any particular vector we choose?
    A vector corresponds only one co-vector , such that . . . A vector space and its dual vector space have same cardinality, that is . . .
    So each vector has a unique covector and that covector acts upon that vector to give one unique number ?

    If that is correct what would be the number for the vector [3,3]?

    As a second (perhaps inconsistent) question ,is their one set of possible covectors for any particular set of vectors or can their be any number of vector/covector set pairings that we can choose arbitrarily?
    Last edited by geordief; July 1st, 2019 at 05:28 AM.
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  27. #26  
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    Quote Originally Posted by trfrm View Post
    A vector corresponds only one co-vector , such that
    But this is only true for metric spaces i.e. for vector spaces with a defined inner product, as I explained in an earlier post. Not all vector spaces are metric spaces
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