For two real vectors $X_1$ and $X_2$, by "linear combination", we mean $aX_1 + bX_2$ for any $a$ and $b$. We use this term while defining vector spaces and related things.
Is there any standard name for such combination: $AX_1 + BX_2$, where $A$ and $B$ are real matrices? Like "Linear matrix combination"? Is there any standard literature where people have investigated the properties of such combination?
'Evaluating' $AX_1$, for all $X_1$ will give the column space of $A $, and similarly for $B$. The sum of these two will give the set of vectors $\{u+v\}$, wher $u,v $ are elements of column space of $A $ and clmn space of $B $ respectively. That is, we get the (algebraic) sum of the two subspaces (also a subspace). Considering the matrices as operators this is also obviously $Im (A)+Im (B) $.
Sum of such subspaces and similar have been and are studied, and will be considered for example in operator theory. Books on Linear algebra may consider this and related questions.