Math 110 - Spring 2001 - Curtin - Final

# Math 110 - Spring 2001 - Curtin - Final - 1

• Test Prep
• 1

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: 11/11/2001 SUN 12:10 FAX 6434330 MOFFITT LIBRARY Math 110 Section 2, Spring 2001 FINAL EXAM B, Curtin Problems will be graded for correctness/ completeness, so be sure to include explanations which justify each step in your arguments and computations. For example, check that the hypotheses of any theorem that you use are satisﬁed. Do not interpret the problems in such a Way that they become trivial—if in doubt, ask. Problems will also be graded for for how well they are written. Ideally, you should use complete sentences to explain your ideas. Of course equations can be included in such a proof. Should it become necessary to leave the room during this exam (eg. ﬁre alarm), this exam and all your work is to remain in the room, face down on your desk. go . Let A be a 2 X 2 real matrix with eigenvalues l and 0 corresponding to respective eigen— vectors (:15) and (:31). Show that A is symmetric. Let T be a linear operator on a ﬁnite dimensional complex inner product space V. Let a: E V. Show that if T is self—adjoint, then T2(.r) = 0 implies that T(:r) = 0. Let V and W denote vector spaces over a ﬁeld F. Let T and U denote nonzero linear transformations from V into W such that R(T) ﬂ R(U) = {0}. Prove that {T, U} is a linearly independent subset of £(V, W). . Prove that the eigenvectors of a complex normal matrix which correspond to distinct eigen— values are orthogonal. Let A be a matrix over a ﬁeld F. Prove that if A is an eigenvalue of A, then /\ is an eigenvalue of At with the same algebraic and geometric multplicities. Let A be an n X n matrix over a ﬁeld F. Prove that dim(span{I, A, A2, ...}) S 17.. Let V denote an inner product space. Fix y, z E V and deﬁne T : V —-} V by T(\$) = (:23, y)z. Show that T is linear and that T* exists. Give an expression for T*(w) involving cc, y, and 2. Let A and B be n X n matrices over a ﬁeld F. Prove that if AB 2 I then BA = I. (Here you are to use the deﬁnition of inverse, just as you did when you did this problem in homework). Let 2 1 A=(1 2)‘ Observe that A is a real symmetric matrix so it is orthogonally equivalent to a diagonal matrix (you don’t have to do anything about this comment). Find a diagonal matrix D and an orthogonal matrix Q such that Q‘IAQ = D. I001 ...
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