In the book "Essential Linear Algebra with Applications", written by Titu Andreescu, there is the following definition:
As you can see, at the bottom of the image appears the symbol $\cdot$ about which nothing is said. Since I was creating notes about the section where this definition is, I decided to "formalize" this definition by doing the following:
Let $F$ be a field. We define an binary operation $\cdot:M_{m,n}(F)\times F^n\to F^m$ such that for $A=[a_{ij}]\in M_{m,n}(F)$ and $X=[x_i]\in {F^n}$ we have that $A\cdot X=Y\in F^m$ in which $Y=[y_i]$ with $y_i=\sum_{k=1}^na_{ik}x_k$. We obtain therefore a map $F^n\to F^m$ which sends $X$ to $A\cdot X$.
Although the definition I made seems to make sense, I'm afraid that this definition does not make sense. Since I intend to continue using the definition above, I would like to know if it is equivalent to that of the book and if it is correct taking into account the already known properties of the multiplication between matrix and vector.
Yes, it is correct. Note that the operation mentioned by Titu Andreescu is simply the product of two matrices, in a case in which the second matrix has a single column.