# Connections between Map Reduce operations and Linear Algebra [closed]

I am interested with Map and Reduce operations in computer science. I would like to do analogies with Mathematics and especially with Linear Algebra.

1. Reduce

First, could we say that "Reduce" operation can be translated as a linear form, ie I start from a vector and I get a scalar from the coordinates of the initial vector ?

But a linear form is by definition linear, is it always the case for Reduction in computer science ?

1. Map

Secondly, could we say that "Map" can be translated as an endomorphism, i.e we start from an initial vector (like a list in computer science) and we get also a vector as output.

If I take "Map" operation, and f a function, I have :

$f[x_{1}, x_{2} ..., x_{n}] = [ f(x_{1}), f(x_{2}), ..., f(x_{n})]$

I could write it like this :

$f[\vec{X}] = f [\sum_{i=1}^{n}\,x_{i}\,\vec{e_{i}}]= \sum_{i=1}^{n}\,x_{i}\,\vec{f(e_{i})}=\sum_{i=1}^{n}\,x_{i}\,F_{ij}\,\vec{e_{j}}=[f(x_{1}), f(x_{2}), ..., f(x_{n})]$

So I get : $f(x_{j})=\sum_{i=1}^{n}\,F_{ij}\,x_{i}$ with $F_{ij}$ called the matrix of change basis of the endomorphism "f"

But I don't get this relation above, well known between a change of basis (Matrix $F_{ij}$) and coordinates into 2 basis :

$x_{j} = F_{ij}\,x'_{i}$

I don't know how to make correct connections between Linear algebra and "Map,Reduce" operations ?

If someone could help to Conceptualize these operations, this would be nice.

Thanks

PS: don't hesitate to transfer this question to mathematics exchange community if necessary.