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 ...
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