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You may as well just title your posts "Vague Title"!
Click "edit" on the first post, then change title at the top.
I don't know either, but here are some thoughts to maybe push things forward a small step.
There is a category Vect whose objects are vector spaces and morphisms are linear maps.
There is a category Set whose objects are sets and morphisms are functions.
There is a forgetful functor
which takes a vector space and maps it to the underlying "set" of vectors.
Whenever you have a forgetful functor laying around, it is usually interesting to look at its adjoint functor which goes the other way
This takes a "set" and gives a vector space. In physics/engineering, you'd say the "vector space spanned by ".
So the beginning of your question could be rephrased to something like "Say we have a set of matrices and form the free vector space ."
I'm not sure where to go from here other than maybe to point out that the determinant is a functor.
Ok. Hopefully an expert can step in now :)
A general nonsense formalism, not in the sense of category theory, to bases and linear independence, has been axiomatized by a Bourbaki member Whitney, under the name "matroid". Do we have an entry matroid ?
Well, just to pick up where Eric left off: suppose is an index set for a collection of vectors in a vector space (for example, could be a vector space of matrices). In other words, suppose we have a function which takes to . In other words, suppose we have a function , where is the underlying set of ( being the functor which forgets vector space structure).
Then, by the free-forgetful adjunction to which Eric refers, the function corresponds uniquely to a linear map .
The collection is linearly independent if is an injection (a monomorphism); equivalently, if the kernel of is trivial.
Could you make this question about the tensor product more explicit?
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