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**Unformatted text preview: **Math 211: Linear Algebra Midterm 1 Spring 2011 Remarks and Statistics Special Notes: a) Textbooks, notes, notecards, calculators, cell phones, electronic devices, etc. are NOT allowed. b) To obtain credit you need to show all your work. Justify all of your claims and arguments fully, and write clearly and legibly. c) Use the backs of the pages if you need extra space. Be sure that you have read all of the instructions before beginning any exam! There were many people who did not abide by the instruction to “justify all of your claims and arguments fully” and consequently did not receive full credit on various problems. This comment applied particularly to problem 2(c). 1 (1) Suppose that A = 1- 2- 2 5 and AB =- 1 2 c 6- 9 d , and let B = h ~ b 1 ~ b 2 ~ b 3 i . (a) Explicitly find ~ b 1 and ~ b 2 . (b) Suppose that ~ b 3 = ~ b 1 + ~ b 2 . (i) Is the matrix equation h ~ b 1 ~ b 2 i x = ~ b 3 consistent? (ii) Explicitly find c and d . (a) There was a lot of unnecessary work done in this problem in general. You can bypass the need for writing out a linear system in terms of variable entries of B by simply remembering that the first column of AB is, by definition, equal to A ~ b 1 . This observation gives you immediately the matrix equation A ~ b 1 =- 1 6 without having to introduce any variables whatsoever. (b) (i) Again, there was a lot of unnecessary work done here by nearly everyone. The matrix equation h ~ b 1 ~ b 2 i x = ~ b 3 is consistent if and only if it is possible to write ~ b 3 as a linear combination of ~ b 1 and ~ b 2 , by definition of the matrix multiplication. It is given in the problem that we can write ~ b 3 = 1 · ~ b 1 + 1 · ~ b 2 , so x = (1 , 1) is automatically a solution. Remark: My goal is to test the new techniques that we’ve acquired in the context of our linear algebra course. If you use techniques for approaching a problem that you have learned elsewhere, you run several risks. First, if you make a mistake it will be nearly impossible for me to award partial credit because I will not necessarily be able to follow your argument or calculation. Second, you might not be developing the necessary new, linear algebraic techniques that we’re hoping to acquire in the course (and will definitely need later, even if not for the present problem). So my premise is that an exam will be testing the new linear algebraic skills from the relevant sections that we’ve covered to that point in the course. If you (successfully) use other types of arguments that you have acquired elsewhere, I will award generous partial credit, but not full credit—mostly to keep us all on the same level playing field, but also to avoid the potential pitfalls mentioned above....

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