What are a, b, u, v, and w? Have you got no information about these?
Simply giving the results of F on 4 separate vectors is not enough to tell whether F is linear or not- it is not enough to determine what F is!
For example, it would be possible to define F to be exactly what is given on those four vectors and 0 for all other vectors. That would NOT be a linear map.
If we are given the result of applying F to four basis vectors, a, b, u, v, for and F is "defined by linearity" for all other vectors (that is, if a, b, u, v are basis vectors, then for any v in , v= pa+ qb+ ru+ sv for some numbers p, q, r, and s, and then F(v)= pF(a)+ qF(b)+ rF(u)+ sF(v), then F is, of course, linear.