
Space of Functions?
What is the set of all functions, and as a consequence the set of all m x n matrices
supposed to look like?
S = {(x₁,x₂,...,x₊)x₁,x₂,...,x₊∊R} is a regular vector space, another notation I've seen for this is:
S = {α ∊ (x₁,x₂,...,x₊)x₁,x₂,...,x₊∊R} which clearly indicates α is a vector and the
operations defined on S are:
+ : S x S → S defined by + : (α,β) ↦ α + β = (x₁,x₂,...,x₊) + (y₁,y₂,...,y₊) = ...
• : S x F → F defined by • : (α,c) ↦ (cα) = c(x₁,x₂,...,x₊) = ...
Now, if we look at the set of functions we see that:
+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
• : S x F → F defined by • : (f,β) ↦ (βf)(x) = βf(x).
so would my set be
S = {(f,g,...,j)f,g,...,j∊R}
or
S = {f ∊ (x₁,x₂,...,x₊)x₁,x₂,...,x₊∊R}?
I can't get the set of functions to follow the format I've used above (Doh)
I don't think either of those sets I've just described make sense either tbh
Then with matrices, if we look at how + & • are defined for functions and notice
that a matrix is just the function
f : (i,j) ↦ A(i,j) = Aij
it seems reasonable that the vector space of matrices is defined in the same way
as the vector space of functions.
I am really not sure, I have a feeling that for matrices the operations are
defined along the lines of:
+ : $\displaystyle F^{m x n}$ x $\displaystyle F^{m x n}$ → $\displaystyle F^{m x n}$ defined by + : (i,j) ↦ (A + B)(i,j) = A(i,j) + B(i,j) = Aij + Bij
• : $\displaystyle F^{m x n}$ x F →$\displaystyle F^{m x n}$ defined by • : ((i,j),β) ↦ (βA)(i,j) = βA(i,j) = βAij
rather than
+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
• : S x F → F defined by • : (f,β) ↦ (βf)(x) = βf(x)
because that notation suggests the m x n dimension characteristic of matrices
where the standard function notation obscures it (I think). Notice the last bit of
notation (the (f,g) ↦ (f + g)(x) stuff for matrices) really doesn't make sense either
so I don't think it can be along these lines.
So, to sum up I'm just asking about the notation describing the vector space of
functions and as a consequence of this the notation for the vector space of all
mxn matrices.
What say you? (Nerd)