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MA554_JAN07

MA554_JAN07 - Math 554(12 Qualifying Exam Heinzer January...

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Math 554 Qualifying Exam Heinzer January 2007 (12) 1. Let F be a field, let n be a positive integer, and let W = F n × n denote the vector space of n × n matrices with entries in F . (i) Let W 0 denote the subspace of W spanned by the matrices C of the form C = AB - BA . What is dim W 0 ? (ii) For A F n × n , define the adjoint matrix adj A F n × n . (iii) If A R 3 × 3 and det A = 2, what is det adj A ? (10) 2. Let Q denote the field of rational numbers. Give an example of a linear op- erator 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. 1

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2 (20) 3. Let A and B in Q n × n be n × n matrices and let I Q n × n denote the identity matrix. (i) State true or false and justify: If A and B are similar over an extension field F of Q , then A and B are similar over Q . (ii) Let M and N be n × n matrices over the polynomial ring Q [ x ]. Define “ M and N are equivalent over Q [ x ]”. (iii) State true or false and justify: Every matrix M Q [ x ] n × n is equivalent to a diagonal matrix. (iv) State true or false and justify: If det( xI - A ) = det( xI - B ), then xI - A and xI - B are equivalent. (v) State true or false and justify: If A and B are similar over Q , then xI - A and xI - B are equivalent in Q [ x ].
3 (14) 4. Let F be a field, let m and n be positive integers and let A F m × n be an m × n matrix.

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MA554_JAN07 - Math 554(12 Qualifying Exam Heinzer January...

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