Functions are vectors
thenumb.at
Functions are vectors
1–10 of 124 posts
Re: Functions are vectors
#2Meditating on the converse statement is also an interesting thought exercise: A vector is (just) a (cached) function (evaluation).
Re: Functions are vectors
#3This 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.
Re: Functions are vectors
#4Meditating on the converse statement is also an interesting thought exercise: A vector is (just) a (cached) function (evaluation).
yep, because of this one can do O(1) sigmoidals in float16 nnets with a 64K-word table.
Re: Functions are vectors
#5I always liked this viewpoint a lot. I'm enjoying with abandon some dusty lectures that Vito Volterra gave in Madrid on differential and integrodifferential equations, while helping also to create Functional Analysis (a Functional being the analogue of a dual vector). He is constantly exploiting this analogy method from finite variable constructions to infinite, also uncountable variables. Even up to showing some embarrassment of being too repetitive with the idea! People in teaching should join and take a peek.
Re: Functions are vectors
#6I always liked this viewpoint a lot. I'm enjoying with abandon some dusty lectures that Vito Volterra gave in Madrid on differential and integrodifferential equations, while helping also to create Functional Analysis (a Functional being the analogue of a dual vector). He is constantly exploiting this analogy method from finite variable constructions to infinite, also uncountable variables. Even up to showing some emb…
Givental uses this viewpoint too in his differential equations class notes. It may soothe students who notice a disjunction between linear independence of functions and that of vectors.
Re: Functions are vectors
#7This 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.
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)
Re: Functions are vectors
#8There’s probably a PhD or two waiting to be earned exploring the relation between Dijkstra/Scholten boolean structures and vector spaces.
Re: Functions are vectors
#9Meditating 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.
Re: Functions are vectors
#10This 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.
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)
"In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y." [0]