MATH 220 Final, Spring 2004

Math 220 Fina} Exam June 3, 2004 Throughout V denotes a finite dimensional space over a field K and T: V— V denotes a linear transformation. 1. (i) Define the minimal polynomial m(x) = mr(x) € K[x] of a linear transformation T: V > V. (ii) Prove the existence and uniqueness of m(x). (iii) Lf m(x) = ag + uxt... +2", prove that T is an isomorphism if and only if ag # 0. (i) Define what is meant by the statement, an element & € K is a characteristic root of T. (ii) Prove that & € K is a characteristic root of T if and only if m(&) = 0, where m(x) is the minimal polynomial of T. (i) Define what is meant by the statement V is a direct sum of the subspaces and Vo. (ii) Prove that every subspace of Vis a direct summand of V. (iii) If the linear transformation T is nilpotent, define the index of nilpotency k of T. and prove the existence of v € V such that the subset {v, TO), .... T'7'(@)} of Vin linearly independent. (iv) Conclude that k < dim V. (i) Let & be a characteristic root of T, v the associated characteristic vector and let vu . Kv > V denote the inclusion map. Prove the existence of a linear transformation p : V + W of vector spaces such that the sequence 0+ Ky »V~“~> W- 0 is exact. Gi) Let S : ¥ - W* be a linear transformation of vector spaces such that Su = 0. Prove the existence of one and only one linear transformation S* : W— W* such that S*p = S. State the primary decomposition Theorem associated with the minimal polynomial m(x) of T. (i) If T is nilpotent, prove that the minimal polynomial of T is of the form m(x) = x‘ where & is the index of nilpotency of T (11) Make a brief account without proofs why there exists a basis of V such that the matrix of T consists of square matrix blocks each having the entry 1 along the lower super diagonal and 0 entries elsewhere. State the Jordan decomposition theorem of a linear transformation T : V+ V. Describe briefly without proofs the Jordan matrix form of a linear transformation.