Exam 1 Fall 2013 - Math 318 Test 1 Oct 7 2013 l 2 3 Tomi...

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Unformatted text preview: Math 318 Test 1 Oct 7, 2013 l. 2. 3. Tomi: Name: Student N0. 45 minutes. No aids are allowed. There are 3 problems and 3 pages in total, each is worth 10 pomts, for a total of 30 points. Problem 1. (3+ 7) (a) For the set of vectors {V1,V2, . . . ,Vk} 111 R”, define the span of {V17V21. . . ,Vk}. k h v’ ’a‘ ’- v" ,— 2 fl»? Mu ‘ I" V e K V a» “w I CI) 6 J 5* (b) For a subspace. V of IR”, show that {V1,V2, V3, . . . ivk_1,v,z€} span V if and only if {v1 —VQ, V2 ~ V3,V3 m V4, . . .,Vk_1 - Vk,Vk}SpaI1V. E?” 3» J 4;. 91:0“; hiat’ra” ’bfgra“; J Lsfiiaa L?! ,s «— as!» ,azgmsgfl, “:2, {vv HM wav ‘ ‘ dd J‘ ‘9‘” a ’ ' I (>i J J.“ {5‘ J"; r I": ('— PQ‘" ) 4 L, v (9- 6’R @7VVC’V, V“. Li J J“ “F” J J“ Problem 2. (E444) (a) Define the notion of open sets of R”. A @906 (ACIRM {5 of!» {j ?a'c’ka- 376M, we”: Mia/“t; 5>0 MW s(s,s)=§§ém“1“5*§”<flc M“. (13) Let U and V be open sets of R”, ShOW that the intersection U H V is open. was) 6 WV- {c} Show that the set S:{(m,y) 6R2 $2+y2 < 1, cc>0} isopen. Problem 3.(5+5) Define the function f : R2 —> R flu" y} m ifm+y%0 I O iffliiryzo (a) Show that f is a continuous function. That is, f is continuous at a for every 21 E R2. w¢w~£r€ww é—w 1% w k ’15} M)#o ” J” fix— fleas“ :waw,ut) Liam/i) 44"“ “Mi ...
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