# Dual Space – The dual of the dual is itself

Good morning, Jakarta. (FYI, I’m on Jakarta now, preparing for our Chinese New Year family celebration :D)

Recently, I found some interesting algebraic properties of a vector space. Some kind of abstract object, however the term is popular among other areas of mathematics. Comments are pleased for correction & other additional information.

First, to make familiar with the ‘muggles’ (#orang awam matematik), let us define some terms:

A field is a set, such that for each the following is hold:

- There exist such that
- There exist such that
- There exist such that
- There exist such that $latex a^{-1} \times a=1$

A vector space over a field is a set, such that for each and the following is hold:

- There exist such that
- There exist such that
- There exist such that

An isomorphism from a set to is a bijection from A to B in which the operations involved in each sets are preserved. Naively, to operate first then transform is the same as to transform first then operate.

A homomorphism from a set to is a mapping from A to B in which the operations involved in each sets are preserved. (no need the mapping to be a bijection)

The set is said to be isomorphic to if we can find any isomorphism from to .

Now let the magic begin. Suppose we have a vector space over a field . Then we can find some homomorphism from to . Moreover, if we collect every single homomorphisms and gather them in a set, namely homomorphism from to . Then we can make an operation such that is a vector space under operation . Next, as we always construct a vector space from arbitrary vector space , we may define as the dual space of .

Well, we now have the structure of dual space. Now, let we have a vector space and its dual . As the is also a vector space, we can construct its dual, let say . Magically, one had proven that is then isomorphic to . 2 isomorphic sets means that the two sets are algebraically equivalent. Then, we can conclude that every vector space is equivalent to the dual of its dual.

What a properties..

Moreover, the concept of duality is not only exist in linear algebra. You may check that the duality may appear on linear programming, graph theory, and so on. I believe that the concept of these things are not different. In abstract they are ‘isomorphic’, but somehow I have not discovered how the may come together as one. That’s all guys. May the magical properties of algebra fulfil your heart. Thanks for your attention. 😀

Enjoy algebra!

wah,ini aku banget~

hehehehe

(*dek ini arti aljabarnya apa sih?)