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The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.

Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is said to be defined over $k$ if $f(V_k)\subset W_k$ and these are elements of $Hom_K(V,W)_k\subset Hom_K(V,W)$. This is a $k$ structure if $W$ is finite dimensional.