If the vector spaces are defined on the same scalar field, then yes.
Let [tex]E=\{e_1,\ e_2,\ldots,\ e_n\}[/tex] be a base of V and [tex]F=\{f_1,\ f_2,\ldots,\ f_n\}[/tex] be a base of W.
We can build the function
[tex]T:V\mapsto W\quad f(e_i)=f_i[/tex]
We can extend T as one may imagine: every [tex]v \in V[/tex] can be written as a linear combination of elements in E:
[tex]v = \displaystyle \sum_{i=1}^n \lambda_i e_i \implies f(v) = \sum_{i=1}^n \lambda_i f(e_i) = \sum_{i=1}^n \lambda_i f_i[/tex]
