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I'm trying to find a way describe in mathematical terms a function that takes one or more inputs but the input and output of the function have the same dimensions or shape.

Let's say $x$ is a vector of length $n$. $y$ is also a vector of length $n$ is there a term that describes a function $f$ where $f(x,y) = z$ and $z$ has the same length as $x$ and $y$. The terms isomorphism and homomorphism have been thrown around but I feel like either these terms are not narrow enough or they are too complicated to understand in this simple case.

Essentially I'm looking for a term (and maybe a branch) that describes relations of this type, the shape of the input matches the shape of the output. And if there is a term that describes the opposite where the shape of the output does not match the shape of the input that would be helpful too!

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Well, it depends. Since you're asking this from a linear algebra point of view, do you want the function to preserve the vector space structure? This is what "homomorphism" means. In this particular case, we also speak of "linear transformations".

For your example, you could simply say that $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a function from the cartesian product of $\mathbb{R}^n$ with itself to $\mathbb{R}^n$. This means exactly that $f$ takes pairs of vectors and outputs a single vector. If you also wanted this function to preserve the linear structure, you'd say that $f$ is an homomorphism or a linear transformation.

By the way, if the domain is the same as the codomain, e.g. $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$, we speak of an endofunction or endomorphism.

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That's perfect, thanks – lukecampbell May 3 '13 at 17:40

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