1.1 1.2 Show that if w is continuous on [0, 1] and jwvdx=0 VveV, ic. p / then w(x)=0 for xe[0, 1]. Show that under suitable assumptions the problem B above can be given the formulation (1.1).

Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T), for every T E L(V, W).

31,37

Q2. Consider the following three functions: f : Z+ x Z+ → Z+ f(m, п) %3D тп g: Z+ → Z h: Rx Z+ → Z+ g(n) = n – 9 h(x, y) = 7y +1 Which 3 of the following statements are true? Only 3 statements are true. A. f is injective (one-to-one). B. f is surjective (onto). C. g is injective (one-to-one). D. g is surjective (onto). E. h is injective (one-to-one). F. h is surjective (onto). G. g is invertible. H. The domain of the composition gof is Z+ I. The codomain of the composition gof is Z

10.2.C. (3rd bullit point). Thanks.