Can someone help me with these? I know these are pretty lengthy but it is midterm season and I cant figure these out/have the time for it TT.

A function is bijective if it is both injective and surjective. Q3. Given a set of vectors S = {V1, ..., Vn} consider the coordinate linear map T : R" - V given by T(X1, . .., In) = XIV1+ 2V2 + ... + n Vn. (a) What is the image of T? (b) What is the rank of T? (c) Prove that T is bijective if and only if S is a basis for V. Q4. We will later show that if V is a finite dimensional vector space, then T : V - V is injective if and only if it is subjective if and only if it is bijection. This is not true for infinite dimensional vector spaces. Consider the following two operation. RS(X1, T2, X2, ...) = (0, X1, X2, ...) and LS(x1, T2, 23, ...) = (22, 13, ...) (a) The function RS is called the right-shift function. Show that it is injective but not surjective. (b) The function LS is called the left-shift function. Show that it is surjective but not injective. Q5. Suppose that V and W are finite dimensional vector space. Prove the following: (a) There exists an injective linear transformation T : V - W if and only if dim(W) 2 dim(V). (b) There exists a surjective linear transformation T : V - W if and only if dim(W) Rank(T2). (b) Prove that if Rank(T") = Rank(Tn+!), then Rank(T" ) = Rank(Tint*) for all finite k > 0. (c) Suppose that T is a nilpotent operator. What can we say about nullity(Tint!) if Rank(T) > 0? (d) Suppose that T is not nilpotent. Show that there exists some operator S = T* such that V = image(S) @ ker(S). (Hint: It is easier to prove an equivalent statement from question 1.)