Consider three vectors spaces U , V, W all over the same eld ]F, with Z1, zz, Z3 being three particular vectors in W. Suppose that S E C(U, W) is an injective linear map, and that {x1,xz,X3} is a set of linearly inde- pendent vectors in U such that 8(X) = zi for i = 1, 2,3. Suppose that T E (V, W) is a surjective linear map, and that {y1,y2,y3} is a set of vectors that span V such that T(y) = Z; for 2' = 1, 2, 3, a) Prove that {21,22, Z3} is a basis for W. b) Find dim(W). c) Show that {1:1, 1:2, 1:3} is a basis for U and that {y1, y2, y;;} is a basis for V