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.

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.
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.
@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 06:13 AM.
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..
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!
Maybe it is too simple, but I have the impression there are basically two kinds of vectors, covariant and contravariant. The distinction is important in differential geometry.
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...
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 socalled metric tensor
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.Are all these vector spaces always infinite and continuous?
In general, infinitedimensional vector spaces are complexvalued i.e. in the above the are complex numbers.
Here's a puzzle for you: given a vector space of complexvalued 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)
Yes, in fact there almost always are. I'll explain in a minute.
Yes you can for the finitedimensional case. Even experienced mathematicians do sometimes.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)
So to your earlier question, using 2dimensional 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?
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
I don't quite get it.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 ?
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
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 spoonfed.
Good luck
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)
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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