Vector Space of Matrices: How to Define and Illustrate?

[sponsoredwalk]

You know the example "The space of functions from a set S to a field F" that's usually
given in a linear algebra text? Well they never give an example of the set they're working in
in detail so I defined the set as:

((S, (S x S, S, +)), ((F, (F x F, F, +')), (F x F, F, °)), (S x F, F, •))

where:

+ : 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).

&

(S, (S x S, S, +)) is an abelian group of vectors (functions),
((F, (F x F, F, +')), (F x F, F, °)) is the field over which the operations take place,
(S x F, F, •) is the operation of scalar multiplication on vectors (functions).

My questions is: How would I illustrate a vector space of matrices akin to the notation
above?

Since a matrix is just the function f : (i,j) ↦ A(i,j) = A_ij (as defined in Hoffman/Kunze anyway!).
I think the function is more generally defined as
f : F^{m x n} x F^{m x n} → F^{m x n}

To translate it into the above language I'm thinking:

((F^{m x n}, (F^{m x n} x F^{m x n}, F^{m x n}, +)), ((F, (F x F, F, +')), (F x F, F, °)), (F^{m x n} x F, F^{m x n}, •))

where

+ : F^{m x n} x F^{m x n} → F^{m x n} defined by + : (i,j) ↦ (A + B)(i,j) = A(i,j) + B(i,j) = A_ij + B_ij

• : F^{m x n} x F → F^{m x n} defined by • : ((i,j),β) ↦ (βA)(i,j) = βA(i,j) = βA_ij

But that seems weird tbh, is it correct?

 

[HallsofIvy]

? You said you were going to define a set but what you define is not simply a set. A set would be something like "{a, b, c}". What would be the set of functions from that to, say, the field fo rational numbers?

 

[sponsoredwalk]

I'm just going by the guidelines I've been given in this post:

https://www.physicsforums.com/showpost.php?p=3105066&postcount=7

Maybe the language was a bit loose, re-read the sentence:

"Well they never give an example of the set they're working in in detail so I defined the set as:"

as:

"Well they never give an example of the sets they're working in in detail so I defined the vector space as:"


and hopefully it will make more sense.