(a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent, hence form a basis of V. (a.1...

Extracted text: (a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent, hence form a basis of V. (a.1 Show that there exist a, ai, ..., an-1 E K so that f"(v)= an-1f"(v) + ... + af(v) + a0v. (a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)} (a.3) Find the characteristic polynomial of f