Vector Spaces are Free Objects

by Musa Al-hassy   Last Updated September 21, 2018 20:20 PM

Warning: I know little linear algebra and my assertions below may all be incorrect. I am interested in lists --i.e., free monoids-- and my interest has led me to [finite-dimensional] vector spaces.


Let 𝓀 be any field, then we can freely equip any [finite] set S with a 𝓀-vector space $F(S) = (S β†’ 𝓀)$, a set of functions where vector addition is performed pointwise, and likewise for the other vector operations. Moreover any function $S β†’ |𝒱|$ from $S$ to the underlying set of a vector space 𝒱 can be extended to a linear operator $F(S) β†’ 𝒱$ that behaves the same on elements of $S$ construed as vectors of $F(S)$.

That is, $F$ is a β€œfree functor”: It constructs the least vector space that contains a (copy of a) given set.

It is known that every vector space π“₯ admits a basis Ξ² and so is isomorphic to $F(Ξ²)$ --which is essentially the dual space of π“₯. That is, every vector space is free; i.e., is in the image of functor $F$.

Of-course we have to choose a basis to realise a vector space as a free object; there is no canonical basis.

( Aside: With this in-hand, we can easily prove equi-dimensional spaces are necessarily isomorphic: That they're equi-dimensional means their basis are in bijection, but free objects are unique up to unique isomorphism and so their F-images must be unique up to unique isomorphism as well. That is, $π“₯ β‰… 𝓀^{dim 𝒱} = 𝓀^{dim 𝒱} β‰… 𝓦$. Neat stuff! )


Questions:

  1. Intuitively why is it that vector spaces admit basis.

    • I'm not interested in a proof.
    • For example, why is it that monoids or rings are not always generated by some set, but vector spaces are. The definition of a vector space does not immediately give rise to a basis.
  2. All [finite dimensional] vector spaces admit basis and so the objects of 𝑽𝑬π‘ͺ, the category of finite dimensional vector spaces and linear operators, are all images of a free functor. (Assuming we have a way to choose basis.)

    • Is there a name for categories whose objects are all isomorphic to an image of a free functor?
    • What other examples of such categories are there?
    • Are there any constructions that produce such categories.
  3. Lists over a set $S$ are isomorphic to $⋃_{n ∈ β„•} Sⁿ$ and provide the free monoid over a given set $S$. Why is it that lists, whence monoids, are far more prevalent than their vector space counterparts within computing science.



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