This is only true if the codomain has the relevant structure for vector operations. Functions are more general than vectors.
Functions are vectors
21–30 of 124 posts
Re: Functions are vectors
#22This touches on the actual definition of a function, which is a mapping between sets where every element of the first set maps to exactly one element of the second set. The problem with using vectors is that vectors aren't as general as sets, so there's functions that can't be expressed using vectors. For example, vectors can't be used to handle undefined values or non-numeric elements.
Functions can’t be used to handle undefined values either. A function is a mapping, f, from one set to another such that each element of the originating set gets mapped to one and only one element of the target set. This, by definition, requires that all values in the originating set get mapped to something in the target set. So there are no undefined values as such.
Function spaces can’t always be viewed as vector spaces because there may be no concept of addition of functions or a concept of scalar multiplication on the functions that behaves well with whatever notion of addition the functions satisfy.
Re: Functions are vectors
#23This is a fascinating take as far as I can follow it, which unfortunately is not that far. But does any of this formal logic help with deriving a function that describes a vector? Because it seems like the greatest inefficiencies and bottlenecks in big data analysis e.g. training networks still boils down to how to find functions that approximate output comparable to expected vectors, whether that's done by symbolic…
You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties (vector addition, scalar multiplication, etc.) and specifically how functions meet the definition, giving you vector spaces of functions (function spaces).
So the idea is that if you two functions, f and g, and a scalar b, then you can do stuff like:
f + g = g + f
b(f + g) = bf + bg
The existence of (-f) so you have:
f + (-f) = 0
Where 0 is the zero function (which also must exist for function spaces).
Re: Functions are vectors
#24This is a fascinating take as far as I can follow it, which unfortunately is not that far. But does any of this formal logic help with deriving a function that describes a vector? Because it seems like the greatest inefficiencies and bottlenecks in big data analysis e.g. training networks still boils down to how to find functions that approximate output comparable to expected vectors, whether that's done by symbolic…
You should look into the pigeonhole principle.
Looked it up. How does that apply to this, or am I lacking the imagination to see it?
Re: Functions are vectors
#25Earlier quoted context omitted.
That's not the definition of a function, what you're describing would be called a bijective function. A simple function that is not bijective and maps to two distinct values would be sqrt(x)
That is the definition of a function, but it's purely a formalism. In reality, and in theoretical math, we are frequently interested in functions that generate multiple values, and it's very easy to represent those in terms of the formal definition: 1. When we want to consider both positive and negative square roots, we can just say that instead of the function sqrt(x): ℝ ⟶ ℝ which always gives a nonnegative real num…
We don’t allow multiple values for a given input of a function. Ever. This is never done. f(x) = x^2 is not invertible as a real valued function of a real variable. On the restricted domain of [0, infinity) it is invertible and that inverse is sqrt(x). If you want to talk about negative roots you use the function -1*sqrt(x). You can construct a new function as you did to encapsulate this fact but one never has a function with multiple values for a given input. I don’t know of any mathematician that thinks of functions as allowing them to have multiple values for a given input. You will not find this in the literature.
Re: Functions are vectors
#26It's easier to see functions as vectors if you first see vectors as functions.
Re: Functions are vectors
#27Meditating on the converse statement is also an interesting thought exercise: A vector is (just) a (cached) function (evaluation).
Caching and evaluation doesn’t make any kind of sense for mathematical functions. They’re just mappings.
Yeah, but it’s true in a sense because an element of a finite-dimensional vector space can be thought of as a function from a finite set into a field.
Re: Functions are vectors
#28I’d probably have titled this something including either the term “linear” or “functional analysis”. Because submitted here, we will first interpret “functions” in the context of a function in computer programming, where the statement is more provocative and thus clickbaity.
The problem is many real world functions and problems are nonlinear. But they may have linear components. For example, a dog can be recognized by its outline and the texture of the fur, pattern of the face, etc within that outline. Deep neural nets solve this by composing vector operations, hence the universal approximation theorem and existence of NNs that can recognize dogs (though I could have picked a better example as dog recognition is not continuous I think).
In the context of computer programming, it is not really a helpful statement to say that functions are vectors. But because of the universal approximation theorem and its relatives you could say that “functions are (can be approximated as) compositions of vector operations”
Re: Functions are vectors
#29This is a fascinating take as far as I can follow it, which unfortunately is not that far. But does any of this formal logic help with deriving a function that describes a vector? Because it seems like the greatest inefficiencies and bottlenecks in big data analysis e.g. training networks still boils down to how to find functions that approximate output comparable to expected vectors, whether that's done by symbolic…
You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties (vector addition, scalar multiplication, etc.) and specifically how functions meet the definition, giving you vector spaces of functions (function spaces). So the idea is that if you two functions, f and g, and a scalar b, then you can do stuff like: f + g = g…
Isn't this just describing a tautology?
I was reading here earlier today about the naming of the constant for light as c, and I had a question which I was too embarrassed to ask. It is this: In e=mc^2, what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant?
Not that I understand a damn thing about that equation. But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial.
This is where I fear I am too stupid to understand the value of this.
Re: Functions are vectors
#30I have never seen these index functions used as a transfinite basis for a vector space. And it seems like the function is not a limit point of finite sequences of basis functions, but some weird transfinite sum with mostly zero entries? Clearly there is no Fourier transformation possible on all functions? I think diagonalization methods would be easy to disprove any useful result.
Even Hilbert spaces are usually just indexed by the ints. And such a basis gives you zero continuity or differential conditions. All the functional analysis I have seen uses some continuity conditions and has some countable basis. Other than that, it is a very useful perspective on functions, and kind of the required start to understanding quantum mechanics formalism.