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This was an example used in my linear algebra class as soon as they started introducing the vector spaces in an abstract sense. I think this post may still be
by azernik 8y ago
This was an example used in my linear algebra class as soon as they started introducing the vector spaces in an abstract sense.
I think this post may still be too wedded to the idea of linear spaces and vectors being arrays of objects - specifically in insisting on decomposing functions like sin and cos to Taylor Series. In fact, you can have a vector space where, in addition to polynomial terms, there are also dimensions for sin(x), tan(x), sin(x - pi), e^x, etc. The fact that you can't enumerate these dimensions, or even describe the set of them until given a set of vectors you're trying to describe, doesn't keep this from being a vector space.
- tzahola 8y agoHm. Interesting. I always viewed real functions as infinite-dimensional vectors in the "canonical" basis, that is, shifted Dirac impulses. I guess it can be transformed into your representation with a change of basis with some handwaving.
- azernik 8y agoWhat you're describing may not be a subspace of the space I'm describing. Mine is definitely a subspace of yours - e.g. you can project functions into sums of shifted Dirac impulses by representing each function dimension as a linear combination of the "Dirac vectors" for that function's values.
- tzahola 8y agoHmmm. I think it’s easy to fix that: Suppose that there’s a function f, that can be written as an infinite sum (integral) of shifted Dirac impulses, but cannot be written in your representation as a sum of those “base functions”. Then simply add a new dimension to your representation that will correspond to f, so that f will be represented as 1 at this new dimension, and zero everywhere else. (In other words: add f to the base functions) Repeat until you have covered every function.