MATH 110 - Spring 2005 - Ribet - Midterm 2 (solution)

MATH 110 - Spring 2005 - Ribet - Midterm 2 (solution) -...

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Unformatted text preview: Math 110 Professor K. A. Ribet Midterm Exam April 5, 2005 This exam was an 80-minute exam. It began at 12:40PM. There were 4 problems, for which the point counts were 18, 10, 10 and 10. The maximum possible score was 48. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Explain your answers in full English sentences as is customary and appropriate. Your paper is your ambassador when it is graded. At the conclusion of the exam, please hand in your paper to your GSI. 1. Label the following statements as TRUE or FALSE, giving a short explanation (e.g., a proof or counterexample). There are six parts to this problem, which continues on page 3. a. A matrix over a field is invertible if and only if it is a product of elementary matrices. This is TRUE. We proved the statement in class, so I won’t say more. b. An n × n matrix with real entries is diagonalizable when considered as an element of M n ( C ) if and only if it is diagonalizable when considered as an element of M n ( R ) . This is simply not true. We could consider, for example, the matrix- 1 1 over R . Its characteristic polynomial is t 2 + 1. Over R , there are no eigenvalues because there are no square roots of- 1. Over C , the polynomial splits into distinct factors, so the matrix is diagonalizable. It’s similar to the diagonal matrix with i and- i on the diagonal. c. If a matrix over F has m rows and n columns, the row rank of the matrix is at most n ....
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This note was uploaded on 07/03/2011 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

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MATH 110 - Spring 2005 - Ribet - Midterm 2 (solution) -...

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