- #1

sponsoredwalk

- 532

- 5

**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?

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?