Tensor product/Functoriality in vector space/Fact/Proof

Proof

(1). This is a special case of fact.

(2). The surjectivity of the mapping

$U\otimes _{K}Z\longrightarrow V\otimes _{K}Z$

is clear, because the ${}v\otimes z$ form a ${}K$-generating system of ${}V\otimes _{K}Z$, and these belong to the image of the mapping.

(3). Because of the injectivity, we may assume that

${}U\subseteq V\,$

is a linear subspace. A basis ${}u_{i}$, ${}i\in I$, of ${}U$ can be extended to a basis ${}u_{i}$, ${}i\in J$, of ${}V$, with ${}I\subseteq J$. Let ${}z_{\ell }$, ${}\ell \in L$, be a basis of ${}Z$. Then, according to fact  (3), the family ${}v_{j}\otimes z_{\ell }$, ${}(j,\ell )\in J\times L$, is a basis of ${}V\otimes Z$. The subset ${}v_{i}\otimes z_{\ell }$, ${}(i,\ell )\in I\times L$, is a basis of ${}U\otimes Z$. Therefore, under

$U\otimes Z\longrightarrow V\otimes Z,$

a basis is mapped to linearly independent elements, and thus this mapping is injective.

To fact