Let V and W be vector spaces with subspaces V1 and W1, respectively if T: V-->W is linear, prove that T(V1) is a subspace of W and that {x in V: T(x) in W1} is a subspace of V.
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.