07win-m1sols - Math 51 Winter 2007 Midterm 1 Solutions...

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Math 51, Winter 2007 Midterm 1 - Solutions February 1, 2007 MIDTERM 1 - SOLUTIONS Complete the following problems. You may use any result from class you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. Please sign the following: “On my honor, I have neither given nor received any aid on this examina- tion. I have furthermore abided by all other aspects of the honor code with respect to this examination.” Name: Signature: The following boxes are strictly for grading purposes. Please do not mark. 1 15 pts 2 10 pts 3 15 pts 4 15 pts 5 15 pts 6 10 pts 7 15 pts Total 95 pts Page 1 of 9
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Math 51, Winter 2007 Midterm 1 - Solutions February 1, 2007 (1) (15 points) Complete each of the following sentences. (a) A collection of vectors -→ v 1 , · · · , -→ v m is defined to be linearly dependent if one of the vectors in the collection is a linear combination of the other vectors in the collection. (You could have also said “if there exists scalars c 1 , · · · , c m —not all zero—so that c 1 -→ v 1 + · · · + c m -→ v m = -→ 0 . ) (b) The dimension of a subspace W is defined to be the number of elements in a basis for W . (c) The book lists 4 properties a matrix must have to be in reduced row echelon form. Two of these properties are (i) the leading entry in every row (i.e., ‘pivot’) is 1 and (ii) if a column contains a pivot, then every other entry in the column is 0. You could have also used (iii) if a row has a pivot, any pivots in rows below are further to the right, or (iv) if every entry in a row is 0, then all rows below also have 0 for every entry.
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