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Math 2270 Exam 2 - University of Utah Fall 2012 Name: 1< This is a 50 minute exam. Please show all your work, as a worked problem is required for full points, and partial credit may be rewarded for some work in the right direction. 1 1. (15 Points) Subspaces Circle the subsets below that are subspaces of Ii: 6he set of all vectors (c) The set of all vectors )he set of all vectors (e) The set of all vectors () ( a 2 \a3 \\ = 1 +a a . 2 with a 3 > 1+a a . 2 with a 3 = 0. with a 3 = 1. I () ( 3 with a a 2 a3 J 2 2. (20 points) For the following matrix, determine the special solutions for the nulispace, provide a basis for the nulispace, and state the di mension of the nulispace. / 1 2 —2 1 \ A= 3 6 —5 4 1 2 0 3 j ( 10u A ( •i1 / L —Z 5 b vc-I— zr2f ro 3 — I 71 / t\ 0 i 0 1 x1, x 1 f, x 1 - $fecl 5o () ( cHf 0 / de%ii’o 3 re ( 3. (20 points) Calculate the rank of the following matrix, provide a basis for the column space, and state the dimension of the column space: B— ,4 t 0 o 7 ro I 4C) 7’ -7 I 13 I /‘t/2Iy rcvL b ( t’àc1 1 k -, \ 1 0 —2 1 0 —1 —3 1 —2 —1 1 —1 0 039 o—i o-0-1-3 0 3 Q Ii 1 0 3 3 —12 1r&cf towL - ( 13 (O Oo \Uo 03-3 o-I /Oo 7/o( -3 (0 100 -I-i 00 000 \oo o-i 3 / ( F 0 enh/I a! 1 p 0 (a/mi7 is 4 3, 4. (10 points) For an 8 x 11 matrix A of rank 5 what are: • dim(C(A)) = • dim(C(A )) T = • dim(N(A))= 5 jj- 3 • dirn(N(A ))= T 5 + 0’ II ç I () N H I II - - If (I D — r - — 11 — Ii 11 II 0 —- C —b-- L- 1 cI’j- I I — co- —Q (oo II j:QGcJ1 II H C 0 CD C C 0 Cl) CD CD C CD CD n (I) C c;1 I’J sil 3 does the following equation have a 6. (10 points) For what value b solution? (Don’t worry about providing the solution, just give the .) 3 necessary value of b 2 3\ 12 5J 3 3 (\2 i 7 4 I L \ 1 / H\131)=( X2 5bcf 3 rI 3 3 /2\ 5). \63J / cfl i 5 / I 3 PrY1 o - A1s I OOQ-7) 7 have -4-• cX —4 -4 U, Ct Ct V > 1- (I ca) — H U — I K r.-) Ji -) -% - U H