MA554_AUG98

MA554_AUG98 - QUALIFYING EXAMINATION August 1998 MATH 554...

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QUALIFYING EXAMINATION August 1998 MATH 554 - Profs. Heinzer/Matsuki Name: (12) 1. Give an example of an inﬁnite dimensional vector space V over the ﬁeld of real numbers R and linear operators S and T on V such that (i) S is onto, but not one-to-one. (ii) T is one-to-one, but not onto. (10) 2. Let Q denote the ﬁeld of rational numbers. Give an example of a linear operator T : Q 3 Q 3 having the property that the only T -invariant subspaces are the whole space and the zero subspace. Explain why your example has this property. (18) 3. Let A and B be n × n matrices over the ﬁeld Q of rational numbers. (i) Deﬁne “ A and B are similar over Q ”. (ii) True or False: “If A and B are similar over the ﬁeld C of complex numbers, then A and B are also similar over Q .” Justify your answer. 1

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(iii) Let M and N be n × n matrices over the polynomial ring Q [ t ]. Deﬁne “ M and N are equivalent over Q [ t ]”. (iv) True or False: “Every matrix M Q [ t ] n × n is equivalent over Q [ t ] to a diagonal matrix.” Justify your answer.
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MA554_AUG98 - QUALIFYING EXAMINATION August 1998 MATH 554...

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