1/?) Time for a short preparatory thread about vector spaces. This is a follow up of my differential form thread: https://twitter.com/LucaAmb/status/1283430683508170754

The goal is to make you realize how weirdly unintuitive vector spaces really are and to gain intuition about the dual space!

2/?) Our intuition of vector spaces is highly influenced by R^n, the space of numeric tuples. Using a coordinate system, we can represent an arbitrary n-dimensional vector space V with a tuple numbers. Good for computation, but it creates a misleading conceptual model!

3/?) R^n comes in fact equipped with a very natural topology and metric! We can definitely say that the two vectors v and w are close to each other, we just need to compare their numerical values right?

True for R^n, completely meaningless for a general V!

4/?) if we use coordinates to map vectors in V to tuples in R^n, we should keep in mind that the resulting metric and topological structure of the tuples is an artifact of the coordinate system! Change coordinates and everything change... except for one thing!

5/?) The real essence of a vector space V is the set of questions you can ask about V whose answer does not depend on the coordinates. What are those questions?

It's simple, there is basically only one type of question you can ask: "Does the vector v lays in the sub-space A?"

6/?) To understand this we need to define sub-spaces. pretty simple, a subspace A is the set of of linear combinations of a set of vectors v_1,...,v_m . A linear combination is an expression of the form:

w = a_1 v_1 + a_2 v_1 + ...

where the coefficients are numbers.

7/?) A vector v lays in a vector space when it can be obtained as linear combination of vectors in the subspace. the question "Does the vector v lays in the sub-space A?" has a binary answer. Yes or no. Nothing else you can ask is meaningful!

8/?) It does not make any sense to ask if a vector v is close to a subspace A. This means that the two geometric representation in the figure are completely equivalent!

The difference is just an artifact of my attempt to represent the vectors on a piece of paper.

9/?) Not understanding the weird geometry of vector spaces blocks you from understanding the nature and importance of the dreaded dual space! The reason of this misunderstanding is that the dual is "trivial" in metric spaces such as R^n.

10/?) Time for some recap. The dual space of V is the space of linear functions from V to R. But does a dual element xi geometrically? In other words, is there a geometrical object in V that can be identified with a dual vector xi?

11/?) Yes! The geometrical element associated with xi is the null space of vectors that are annihilated by xi: {v | xi(v) = 0}.

Dual vectors are called forms. Makes sense! form means shape! Forms are like the toddler toys that only let one shape pass and block everything else!

12/?) In the toddler toy analogy, the form is a hole in the box and the null space is the set of all blocks that do not fit in that shape.

Going back to vectors, why don't instead associate the form with all the vectors that "fit" (give non-zero result)?

13/?) Simple! Because those vectors that "fit" do not form a subspace! Linear combination of those vectors can for sure be in the null space! It's not a nice linear object!

14/?) The null space of a form is a (n-1) subspace of V that is associated with the form xi. If you read my previous thread, you'll know that this picture is almost complete but not quite since scaling xi by a number doesn't change the null space.

15/?) More exactly, the geometric element is a scaled and signed null space. You can interpret it as a stack of null spaces "stacked along a direction of growth". If the form is a gradient of a function, this stack gives the height levels of the function.

16/?) This is what I explained in the previous thread. It is loosely right but potentially misleading, a function on a manifold does not have a well-defined direction of growth!

17/?) The only well defined thing is the scaled null space (space of constancy). A vector can be or not in this subspace but it is not meaningful to ask how close the vector is from it. This means that you can't isolate an orthogonal direction of maximal growth!

18/?) In vector calculus the gradient is a vector giving the direction of maximal growth. This is possible because the metric allows to pick out a well-defined direction of orthogonality from the null space of the gradient form.

19/?) In our toddler toy analogy, the metric allows you to associate a block (well a scaled signed block, the sell them at prenatal...) to each hole.

In adults terms, it gives you a geometrically meaningful isomorphism between vectors and forms.

20/?) More generally, a metric allows you to associate (n-1) spaces to their orthogonal vectors. Each form is then associated to a unique vector... it points somewhere now! This implies that V and its dual become completely equivalent!

21/?) However, in a smooth manifold there isn't a natural metric. this implies that functions f do not have directions of maximal growth! They only have the constancy spaces given by the gradient form! That's why forms and duals are super important in differential geometry!

22/?) Once again, there would be much more to say but it is time to stop. If you enjoyed the thread please like and retweet! Much more geometry will be coming soon! Stay tuned!

A reminder for people who could be interested in this new thread

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