Notation and name for this function?

Question

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?

Answer 1

In some contexts, when direct products are defined (e.g. vector spaces, groups, modules...) given two maps $$f: A \to B \\ g : C \to D$$ one defines $$f \times g : A \times C \to B \times D$$ as $(a,c) \mapsto (f(a), g(c))$.

Sometimes (when you work with modules or vector spaces) products are denoted with the symbol $\oplus$ and are called direct sums. This is your case: usually such maps are denoted by $f \oplus g$. But this is just a matter of taste.