Answer Happy

1. Let V. W be vector spaces, with zero vectors denoted Oy and Ow, respectively. Let T:VW be a linear transformation (a) Prove that T(Ov) = 0w. (b) Let im(T):= {we W: there exists v € V such that T(v) = w}. Prove that im(T) is a subspace of W. (e) Let S be a subspace of W with S Cim(T). Prove that T-'(S):= {v € V:T(v) € S} is a subspace of V. (Note that "T-(S)" is just a symbol we're using, and we're not claiming that I has an inverse function.) (d) Let ker(T):= {v € V: T(v) = Ow}. Prove that if ker(T) = {0v), then for each w e im(T), there is a unique v E V with T(v) = w. (Hint: you could think about proving the contrapositive of this statement. Recall that the contrapositive of the statement " AB" is the equivalent statement "not B not A".) (e) Prove that if ker(T) {Ov}, and S is a subspace of W with S Cim(T) and dim(S) = n, then dim(T-'(S)) = n.