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Unformatted text preview: MATH 115A  Practice Final Exam Paul Skoufranis November 19, 2011 Instructions: This is a practice exam for the final examination of MATH 115A that would be similar to the final examination I would give if I were teaching the course. This test may or may not be an accurate representation of the actual final examination for this course. You should follow the following rules when preparing with this midterm: 1. This exam is meant to be 180 minutes long. Moreover, this exam is meant to be difficult and theoretical. 2. Treat this as the actual final examination. Find time to sit for 180 minutes and go through this as you would the actual midterm. 3. Do not use a calculator or notes as these will not be permitted on the midterm (and the calculator will not help you). 4. Manage your time. There are 12 questions (plus a bonus) so you have about 15 minutes per question. However, these questions are not evenly weighted. 5. When answering a problem, consider how many points the problem is worth and how much space is provided. This can give a clue to how much effort the problem requires, the amount of time you should spend on the problem, the amount of writing expected, and the amount of detail required in your solution. 6. If you finish or are stuck on problems, take time to check your answers. 7. Take your time on each question you know how to do and make sure you give a complete answer. 8. If you get stuck on a question, move on and come back to it if you have time. All right then... Geronimo! Question 1) a) Let V be a vector space over a field F . Define what it means for a subset W V to be a subspace of V . (State the definition given in class.) (2 marks) b) Let W = { A M 3 3 ( C )  det ( A ) = 0 } . Is W a vector space over C with the usual matrix opera tions? Prove your answer. (4 marks) c) Let V be a vector space over a field F and let W and Z be subspaces of V . Is W Z necessarily a subspace of V ? Prove your answer. (4 marks) 2 Question 2) Recall that P 3 ( R ) is the vector space over R of all polynomials with degree at most three. Letof all polynomials with degree at most three....
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 Fall '08
 Kong,L
 Math

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