MATH 220 Final, Spring 1997

MATHEMATICS 220 Time: 2 hrs. (2 Semester, 1996-97) FINAL EXAMINATION Throughout, V denotes a finite dimensional vector space over a field F. 1. (a) Suppose that A is an orthogonal matrix . Find lAl. (b) Let B be a 2x2 matrix over € such that B242B=31. Find |B|, and show that B is diagonalizable ( 10 pts. ) . (a) Let W be a subspace of a vector space V, and suppose that the set of cosets {v.tW, votW, .... Vit W} is linearly independent in the quotient space V/W. Show that the set of vectors{V1 , V2, «++» Va }in V is linearly independent. (b) Give an example to show that the converse of part (a) is false. (10 pts.) 3, Let T bea linear operator on R? which is represented relative to basis of R® by -1 3 0 A=|0 2 0 2 1-1 _ Find the characteristic values and the corresponding characteristic spaces for T. Deduce that T is not diagonalizable. _ Use the Primary Decomposition Theorem to write R? as a direct sum of T-invariant subspaces R? =W, © Wz. Show that W; and W2 are T-cyclic subspaces in this case. . Find the Rational form of A . Find the Jordan form of A. ( 18 pts) 4, Show that all 5x5 complex matrices with characteristic polynomial f(x)= (x-1)"(x+2), and minimal polynomial m(x)= (x-1)°(x+2), are similar. (9 pts) 5, Let A be an nxn matrix over (. Prove that if every characteristic value of A is real, then A is similar to « matrix with real entries. (9 pts) 6. Let T be a diagonalizable linear operator on V where dim V= n. Prove that T has a cyclic vector <> T has n distinct eigenvalues. (10 pts.) 7. (a) Let T be a linear operator on areal inner product space V such that T’=-T. Show that ( T(v), v)=0 for all v inV. (b) Give an example of a linear operator on a real inner product space V such that «:T(v), v)=0 for every v in V but T # 0. ( 12 pts.) . (a) Prove that characteristic vectors corresponding to distinct characteristic values of a normal operator ‘on a finite dimensional inner product space are orthogonal. (b) Let V be a complex inner product space, and let T be a linear operator on V. Prove that T is self-adjoint <= ( T(v), v) is real for all v in V. ( 12 pta) 9. Answer TRUE or FALSE only (2 points for each correct answer, 0 for no answer, and -1 for each wrong answer). a. If Eis a projection on V, then E is diagonalizable. b. If T isa linear operator on a finite dimensional inner product space such that (T(u), T(v)) = 0, for all u,v in V, then T = 0. c. If U and W are subspaces of a finite dimensional inner product space, then (U+W)" = UAW". d. Let T be a Hermitian operator on a complex inner product space V. Then T can be represented by a diagonal matix relative to an orthonormal: basis of V consisting of characteristic vectors. e. If A is an nxn matrix over F with minimal polynomial p(x) = x*-x, then A is invertible. f. If Vis a 2-dimensional vector space over F, and T is a linear operator on V, then the Cyclic Decomposition Theorem gives only one decomposition of V into T-cyclic subspaces. (12 pts.)