1. Let V be a real vector space. We know that V has an additive identity, called 0. Suppose there is a vector u ∈ V with the property that u+v=vforeveryv∈V. Showthatu=0. (Thisshowsthatarealvector space can only have one additive identity.)
2. Let V be a real vector space, and suppose that U1 and U2 are both subspaces of V . The intersection of U1 and U2, written U1 ∩ U2, is the setofallvectorsv∈V suchthatv∈U1 andv∈U2. ShowthatU1∩U2 isa subspace of V .
3. Let V be a real vector space, and suppose that U1 and U2 are both subspaces of V . The union of U1 and U2, written U1 ∪ U2, is the set of all vectorsv∈V suchthatv∈U1 orv∈U2.IsU1∪U2 necessarilyasubspaceof V ? If yes, prove it, if not, describe a counterexample.
4. Suppose that U, V , and W are all real vector spaces, and that T1 : U → V and T2 : V → W are both linear transformation. The composition T2 ? T1 is the function from U to W defined by (T2 ? T1)(u) = T2(T1(u)), for all u ∈ U . Show that T2 ? T1 is also a linear transformation.
5. Consider a linear transformation T : Rn → Rm. Let G be the subset of Rn+m, consisting of all points (x1, . . . xn, xn+1, . . . xn+m) such that T(x1,...xn) = (xn+1,...xn+m). Show that G is a subspace of Rn+m.