a,b,c

4. Consider three vectors spaces U, V, W all over the same field F, with 21, 22, 23 being three particular vectors in W. Suppose that SE C(U, W) is an injective linear map, and that {x1, X2,x3} is a set of linearly inde- pendent vectors in U such that S(x;) = z; for i = 1,2,3. Suppose that T E C(V, W) is a surjective linear map, and that {y1, y2, ys} is a set of vectors that span V such that T(y,) = z; for i = 1, 2,3, a) Prove that {21, 22, 23} is a basis for W. b) Find dim(W). c) Show that {X1, X2, x3] is a basis for U and that {y1, y2, y3) is a basis for V