Suppose that U and V are finite-dimensional vector spaces and that 
Prove that 
.
this is your statement.
you have introduced U and V but not W.
so, ST is not possible with out knowing about W.
so, i ask you to repost the question.
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U,V are finite dimensional.
so, S is a linear transformation from a finite dimensional vector space to another vector space W over the same field. so, we are not given whether W is finite or infinite dimensional.
not even necessary.
we can easily follow that the range of S is finite dimensional.
as it is given ST is possible.
from this it is easy to follow that if the matrix of S is of order m*n then the matrix of T is n*p.
thus the matrix of ST is of order m*p.
suppose the matrix of S is A and T is B, then
if dim null space of S is n-r1, and the dimension of null space of T is p-r2
then we follow that A has r1 linearly independent columns and B has r2 linearly independent columns.but AB has r
min{r1,r2} linearly independent columns.
from this , it follows that dim null ST
dim null S+dim nullT