What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon based on context.
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1See http://en.wikipedia.org/wiki/Distributive_property,http://en.wikipedia.org/wiki/Homomorphism, http://mathworld.wolfram.com/LinearTransformation.html. – Apr 07 '14 at 23:03
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Homomorphism refers to mappings $f \,: X \to Y$ that preserve some kind of structure. Which structure that is in particular depends on the context, or must explicitly be specified. For example, if $f$ is a mapping from group $X$ to group $Y$, then saying "$f$ is an homomorphism" means that $f$ is compatible with the group structure, i.e. that $f(a\cdot b) = f(a)\cdot f(b)$. If $X$ and $Y$ were fields, it would mean that $f(a+b) = f(a) + f(b)$ as well as $f(ab) = f(a)f(b)$. You are correct that for the case of vector spaces, homomorphism means the same as linear.
Distributivity refers to a pair of two operations $u,v \,:\, X\times X \to X$. If $u(x,v(y,z)) = v(u(x,y),u(x,z))$ then $u$ distributes over $v$. Note that you're dealing with mappings of arity two here, instead of arity one as in the case of homomorphisms, and with mappings of a set $X$ to itself, instead of to a different set $Y$.
But yeah, for two operations $u,v$ where $(X,v)$ is a structure of type V (e.g. a group), saying that $x \mapsto u(v,x)$ is a $V$-homomorphism for every $x$ would be the same as saying $u$ distributes over $v$. In the case of a field $F$, that comes down to saying that multiplication with a fixed element $a$, i.e. the map $x \mapsto a\cdot x$, is a group homomorphism from $(F,+)$ to itself. But whether that way of expressing things would be an improvement is another question...
fgp
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A homomorphism is typically a transformation which preserves some sort of structure in an algebraic object: $$\phi(a*b)=\phi(a)\cdot \phi(b)$$
One operation f is set to distribute over another g if $$f(c,g(a,b))=g(f(c,a),f(c,b))$$
A linear transformation (very different) is anything that distributes over both addition and scalar multiplication, usually in a vector space: $$L(c_1x_1+c_2x_2)=c_1L(x_1)+c_2L(x_2)$$
user140943
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It's quite common to refer to linear transformations as vector space homomorphisms, or if they are invertible as vector space isomorphisms. That's, after all, why we call vector spaces isomorphic if there exists an invertible linear transform between them. – fgp Apr 07 '14 at 23:30
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But if we take an operator as a function and make everything live in the same space, then distributivity seems like it a homomorphism. Likewise for the linear transformation. I do not see how they are all that different. To me, only the implied "worlds of operation" change with the nomenclature. – kevin Apr 08 '14 at 07:24