I was wondering if there is a name of the function that maps $(x_1, x_2,..., x_n)$ to $(x_1, x_2, ..., x_n, 0, 0, ..., 0)$? I know when we do it the other way around it's called the projection. And when there is no 0's it's called the identity map. Is there a special name for this one? Thanks!
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I've usually seen maps like this referred to as "the inclusion map". It is often used whenever we identify some object as a subspace of a larger one.
Maps of this kind are injective and linear, so give a well defined injective linear map (or more generally, a homomorphism of modules). If we have such an inclusion $f:U\rightarrow V$ where $U$ and $V$ are groups (or vector spaces, or $R$ modules) then by the isomorphism theorem of your choice, we have $U \cong f(U)$ as groups (or vector spaces, or $R$ modules) which is exactly what we would expect, and just agrees with the fact that we expect the original object $U$ to be somehow embedded in the larger one $V$.