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Unformatted text preview: FINAL EXAM COMMENTS AND SOLUTIONS 1. Comments 1. This was problem 5 on Homework 3. 2. This is a variation on problem 5 of Homework 3. (We have seen operations with this property before; see e.g. problem 2 on Homework 3.) 3. This is like problems 1-4 on Homework 3 and problem 2 on Midterm 1. 4. This is like problems 1-4 on Homework 3 and problem 2 on Midterm 1. 5. This is like problems 1-3 on Homework 7. [The idea: it is required that f be invertible, but that f not be its own inverse. From there you can just look for examples.] 6. This was discussed in class on November 21, and is related to problem 1 on Homework 10. 7. This was discussed in the last week of class: ax + by = c has infinitely many solutions if and only if c is a multiple of gcd( a,b ). 8. This could be done using the Euclidean algorithm (done in class several times) or by any other method you can think of. 9. This was problem 3 on Midterm 2. 10. This is related to problem 5 on Homework 6, and problem 2 on Midterm 2....
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- Fall '07