ok to show that T(V) is a subspace of W, we need to check the following:

1)0 vector is in T(V)

2)for some vectors T(u) and T(v) in T(V), T(u)+T(v) lies in T(V)

3) for some scalar c and vector x in T(V), cx lies in T(V)

condition 1 should be pretty trivial.

for 2, let x and y be two vectors in W that is mapped onto by u and v in V1, so T(u)=x and T(v)=y (note that T(u) and T(v) are also in T(V)). but by linearity, T(u)+T(v)=T(u+v) which is in W since V1 is a subspace of V (this tells us that whenever u and v are in V1, so is u+v). so we have shown that for T(u) and T(v) in T(V), that T(u)+T(v) also lies in T(V).

other conditions should be similar, and involve using the property of linearity.