m1bluekey.pdf

Math 214 ? Midterm 1 Blue Version Answers by Michael Cap Khoury Problem 1: Let d, s, m be the numbers of sacks of Dragon, Serenity, and Miracle Tea that Datura buys, respectively. d+s+m=9 2d+3s+5m=25 25d+30s+20m=230 This system has a unique solution (d, s, m) = (6, 1, 2). Datura bought six sacks of Dragon Tea, one sack of Serenity Tea, and two sacks of Miracle Tea. Problem 2: We use the projection formula. Let L be the line spanned by ? ? 3 4 12 ? ?. projL ? ? ?1 1 1 ? ?= ? ? ?1 1 1 ? ?· ? ? 3 4 12 ? ? ? ? 3 4 12 ? ?· ? ? 3 4 12 ? ? ? ? 3 4 12 ? ?= 13169 ? ? 3 4 12 ? ?= ? ? 3/13 4/13 12/13 ? ? Problem 3: There are infinitely many solutions. What all solutions have in common is that each column consists of cnumbers in arithmetic progression and that the five columns are not all a multiple of a common vector. An example follows. ? ?? ? 1 1 1 1 1 1 2 3 4 5 1 3 5 7 9 1 4 7 10 13 ? ?? ? Problem 4: Since T is compatible with scalar multiples, T parenleftbigg 1 0 parenrightbigg = 13 ? ? 21 15 12 ? ?= ? ? 7 5 4 ? ? T parenleftbigg 0 1 parenrightbigg = 12 ? ? 10 12 14 ? ?= ? ? 5 6 7 ? ?. This gives the matrix representing T, column-by-column: ? ? 7 5 5 6 ?4 7 ? ?. 1 Problem 5: So many possibilities... B1= ?? ? ? ? 1 0 0 ? ?, ? ? 0 1 0 ? ?, ? ? 0 0 1 ? ? ?? ? B2= ?? ? ? ? 1 1 2 ? ?, ? ? 1 2 1 ? ?, ? ? 2 1 1 ? ? ?? ? B3= ?? ? ?? ? ?? 91 423? ? ??, ? ?? 019? 0 ? ??, ? ? pi 6 1 ? ? ?? ? ?? Problem 6: First, we need the rref of A. rref(A)= ? ? 1 0 ?2 5 0 0 1 1 ?1 0 0 0 0 0 1 ? ?. a) The solutions of AxM =0M have the form ? ?? ?? ? 2r?5s ?r+s r s 0 ? ?? ?? ? ; a basis for kerA is ?? ?? ??? ? ?? ?? ? 2 ?1 1 0 0 ? ?? ?? ? , ? ?? ?? ? ?5 1 0 1 0 ? ?? ?? ? ?? ?? ???. b) The third and fourth columns are redundant. A basis of im A is braceleftBigg? ? 1 0 1 ? ?, ? ? 4 1 5 ? ?, ? ? 1 0 0 ? ? bracerightBigg . (Indeed the image is all of R3, so any basis of R3 will do.) c) The rank is 3. d) The nullity is 2. Problem 7: a) C =AB= ? ? ?10 10 3 16 ?5 3 ?5 2 1 ? ?. b) A?1= ? ? 1 ?4 ?15 0 1 3 0 0 1 ? ?. c) B?1= ? ? 1 0 0 ?1 1 0 7 ?2 1 ? ?. d) C?1=B?1A?1= ? ? 1 0 0 ?1 1 0 7 ?2 1 ? ? ? ? 1 ?4 ?15 0 1 3 0 0 1 ? ?= ? ? 1 ?4 ?15 ?1 5 18 7 ?30 ?110 ? ? Problem 8: a) False b) True c) True d) False e) True f) True g) True h) True i) False j) True 2 /home/cap/.TeXmacs/system/tmp/tmp_1087425880.ps