MATH
Math 110 - Spring 2000 - Brown - Final

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

• Test Prep
• 2

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

• Fall '08
• GUREVITCH

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern