Let $A_i:V_i \to W_i, i \in \{1,...,n\}$ be linear maps. Then the tensor product of the these maps is defined to be the linear map $$ A_1 \otimes\cdots \otimes A_n: V_1 \otimes\cdots \otimes V_n \to W_1 \otimes\cdots \otimes W_n, \\ v_1 \otimes\cdots \otimes v_n \mapsto (A_1 v_1)\otimes\cdots \otimes (A_n v_n) $$ Now my question is whether there is a direct sum counterpart to this definition, sth like $$ A_1 \oplus \cdots \oplus A_n: V_1 \oplus \cdots \oplus V_n \to W_1 \oplus \cdots \oplus W_n, \\ v_1 + \cdots + v_n \mapsto A_1 v_1+ \cdots + A_n v_n $$ and whether these definitions are somehow limited by the dimensionality of the space.
Thanks a lot in advance!