Math 110 - Spring 2000 - Brown - Final

Math 110 - Spring 2000 - Brown - Final - FRI 17:42 FAX...

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Unformatted text preview: 03/23/2001 FRI 17:42 FAX 6434330 MOFFITT LIBRARY 001 Math 110 - Final Exam Spring 2000 — Nate Brown 1) (10pts) Assume dim(V) : 6, U C V is a subspace and dim(U) = 4. Prove that there exist one dimensional subspaces U1, U2 C V such that V = U EB U1 33 Hg. 2) (10pts) Assume T E C(V) is invertible and {v1,... ,vk} C V is a lin- early independent set of vectors. Prove that {T(n1), . . . ,T(-uk)} is also linearly independent . 3) (lOpts) Let < -,- > be the dot product on C2 (i.e. < (331,311), (3:2,y2) >: 3215+ 3,123) and T E £(C2) be defined by T(:r,y) = + 215$ +2311). Write down a formula for the adjoint of T with respect to < -, - >. (It does not suffice to just write down some matrix!) 4) (10pts) Assume T E [3032013) has minimal polynomial (z * 1)(z + 4)? Find the matrix of T in Jordan form. 5) Let U, V be vector spaces with di-rri(U) > d‘im(V). a) (lOpts) Prove that if T E £(U, V) then T is not injective. b) (10pts) Construct some T E £(U, V) which is surjectivei a, b # ia+3c —b+u! 6) DefineTE£(M2(C)) byT( c d)_( ic 2d a) (10pts) Find all the eigenvalues of T. (Hint: Consider the matrix of T with respect to the canonical basis.) b) (10pts) Compute the characteristic polynomial of T. c) (lOpts) What is the dimension of the generalized eigenspacc corresponding to the eigenvalue i? d) (10pts) Find det(T). 7)a) (lOpts) Construct an operator T E £(M2 ((3)) whose minimal polyno- mial is z(z w 3)2 and generalized eigenspace corresponding to 0 is one dimen— sional. (Cive both a formula for the operator and it’s matrix in Jordan form.) b) (10pts) Construct an operator 5' E £(M;(<C)) whose minimal polynomial is 2(2 7 3)2 and generalized eigenspace corresponding to O is two dimensional. (Give both a formula for the operator and it’s matrix in Jordan form.) 03/23/2001 FRI 17:42 FAX 6434330 MOFFITT LIBRARY 002 8) For each A 6 IF define TA 6 £(Pm(IF),lF] by Tim) 21200. a) (Spts) Prove that ker(T,\) = ker(TS) if and only if A = X. (Hint: consider the polynornials 107(2) : z — 7.] b) (lets) Prove that dim(ker(TA)) 2 m. c) (153th Let U C PmUF) be the one dimensional subspace spanned by the polynomial 10(2) 2 2. Prove that for every nonzero A E IF there exists a Subspace UA c 79mm) such that 2') U,\ = US if and only ifA = Z\ and a) 73mm = U EB U ,\ for each (nonzero) A. 9) Let Bl : {(1,0,0,0}, (0,1,0,0), (0,0,1,0), (0,0,0,1)} be the canonical basis ofC‘, 32 = {(1,0,0,0), (1,1,0,0), (0, 0,1,0), (u,0,1,1)} and T E 130134) be defined by Tm, b,c,d) = ((1 —— as + (21' + 1)b, —z‘a + (i + 2)b,2c,c). a) (Spins) Compute M(T,Bl, Bl). b) (10pts) Compute the two change of basis matrices M (I , Bl, Ba) and M(Ii B2; B1)' 1 i 0 0 “i 2 0 0 c) (opts) Use part b) to show that M(T, 32,132) = 0 0 1 1 0 O 1 1 d) (lOpts) Prove that T is not normal with reapect the canonical dot product on CC“. e) (10pts) Prove that there exists a basis of (0“ consisting of eigenvectors of T. f) (10pts) Construct an inner product on (C‘1 such that T is normal with respect to that inner product. ...
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