# sol04 - Math 115 Final Exam 2 Page 2 of 13 Name: [6} 1....

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Unformatted text preview: Math 115 Final Exam 2 Page 2 of 13 Name: [6} 1. Find the general solution to the system 121+ 2:22—3:03 + 4.134: —5 321+ 2.1172 +373 — 8114 = 3 'Iz—3‘1—5 {2—37'0’ [12 1'8’3/QZ-/?{ 004-128 [2—3 %- {#3412061 llq‘ez[001*32 001-3?— 0231;; lvz'b Wu :5, ; ﬁgs+51§ +/ 15: 3t-P; Math 115 Final Exam Page 3 of 13 Name: 2. (a) Give the deﬁnition of the dimension of a vector space V (i.e. deﬁne dim(V)). [8] . 52% ( V) = W y We? om ‘M 74d MN) (b)LetH:span i )}.Finddim(H).:-# {WM 4 1 4 mm? “A, 319/ Math 115 Final Exam Page 4 of 13 Name: 2 3 —3 3. Let A = i 4 1 [10] —3 —7 2 (a) Find a basis for the null space of A and the row space of A Le. ﬁnd a basis for N A and RA). 2- -3 0 Ryan; I (I I O I “sf I o “ z. 3 ‘3 0 if“? 13 ~? 2, 0 -5 4 a 0 725*”?! “1,0 f—‘fz lo-zb li’;§%&o,lo"oiao ESKOWLOOOO 0000 Wary; (b) Find the rank of A. and the nullity of A (Le. ﬁnd p(A) and (I) A)). ( . 5M): paw/2,4) :Q wwwm 014* / Math 115 Final Exam ZOO Page 5 of 13 Name: 4. (a) Let 2 2 1+ 2'. Find 215. [8] r: E r (b) Le! 21 = 3 +2" and 22 = 2 — 33‘. Express § and W2 intheform a +5»: 2 ‘ __ _ , . .1_ __§J— :. glib : Orsgz : .——-——-——é+ “L +3b '32» aJ' 214531 x; . I :- 3-H“. :, 3. +_LL L ,3 u 23 gig-2’ _- (3+1: >(Z ’3 L) I t . 'L 3 C3—- ?L+’&L "'3" : 2+3 -:h' 37-9.,” Math 115 Final Exam “2,00% Page 6 of 13 Name: 5. Le! 31 = {1,52,332} and let 82 z '{1 — x,1+ 31,1 + as + x2}. Find T314132, the transition matrix from Bl to 3;. Express 11(3) = 2 — 656 + 4502 in terms of Ba. [10] '- Clufc: Q—éxwfx‘“: ﬁ‘K/a) -é_(/m)+m+14xL) : CFC/7c — 6 ~61: +‘{+Vx+L/>¢L- : Zﬂéx 4/"‘le,1’ w“ ‘ . J Math 115 Final Exam 200 ("f Page 7 of 13 Name: 1 6. LetA 2 1 [121 3 (a) Find the eigenvalues ofA. MK/q' ‘ﬂ) = O 3— l 1 H , — J : )OAQ‘Ai )‘O G) (“E/DJZA3-A) O COW MNH 0. .7. 3% Z _ (=3 (3%) [(27081) *2] “0 (=> (3 7x) C 4) ~—5”A+ A2 ~33 =’ 0 (=3 (3—A§f,\1—D\+‘/3:O C9. L3”)Q(/\”‘/)(x\-r) :O /\"3/ Li / (b) Is A diagonalizabie? Explain. If so, ﬁnd a diagoria] matrix D and a matrix P such that A = PDP". (NOTE: It is not necessaxy to compute P'l.) V I / WU . M 5 1/7" Via“: §:01;0 Ol/D Di!)0 3 o—iio“ooa0"‘0010 0200 OIOO 600C) Math 115 Final Exam Page 8 of 13 Name: a 1 0 5 fully justiﬁed. [81 yrﬂ/‘W am. acme/b A—AI: Z'AJDLA) 5‘: 0 / 0 cg/é—aym 0 1 o 942/ _ [0 5'4, 0 A2,; v we)?! 0 0 0 it“; ’7. Find all the values (if a. and I; such that the matrix ( ) is diagonalizabte. Your answer must be 29,17 51," a"; I 0] 7“in seas 000 12;:—.\$ ._—._.,.~ as vaﬁré we fate/7% W[07M/7fj7 WQWWuéyér-MI 1% mlé [549% Math 115 Final Exam 0 Page 9 of 13 Name: 8. Suppose A is a 4 X 6 matrix, Show that rank (A) g 4 and that nullity(A) 2 2. [5] am aha/MAO ,4de a ‘ M Wr‘mumz‘iqudm 6‘“ “32¢ WWW qmwabmﬁrcry ,’,62mdc(/4)é¥ mm QWMM WM) = (a / WM): é” WMO“) a/kd W(/4) é-Lf (WWW-47‘ e19 é % (A) Qé~%m%§m%9 9. Suppose that A is an eigenvalue of A. Show that A” is an eigenvalue of A2. 51%)”;0 MM ammdpfuu med/(MISOW Ai=k§ wan mu; cm m «674150: A‘ Azik/Hxi A13: WWW), A135 : (ERMA ES am uﬂwﬂﬂﬁx Li: Ali gAz‘ ‘ ‘Mw tﬂl \ Math 115 Final Exam Z00 Page 10 of 13 Name: 10. Let Vbe the vector space ofall 2x2 real matrices. Are the vectors {( _1 > , ( 0 _3 ) , ( 4 1 )} lincaﬂy independent? Explain. Do they span V? Expiain. [10] Math 115 Final Exam ZOO Page 11 of 13 Name: 1 :13 :/#© 1 (L 2 U a 11. Find conditions onasuch thatthe vectors ( 2') , ( —1 ) and ( 2 ) are linearly independent. 1 [8] F a. Math 115 Final Exam Page 12 of 13 Name: 12. Suppose 13,113 E R3 are linearly independent. Let H = span {171,113}. Let U} E R3, but 175 Q H. (a) Show that {13, 113,651} are linearly independent. (Qt/we (91601 apm‘b‘me W -‘ ~%_1{«7,,V v) w M wax/AZ; W [10] ‘wfﬂig: ﬂig+ﬁzg+ﬁ5g 1 3 M W-ﬂ'vfaﬁ fax '144 h «(1.4. nﬂ‘f V65;S M Ugh (pm/3W, Wm kjr0 «M JIZJ #0 Difkéew/KWW w m ...
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## This note was uploaded on 09/17/2011 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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sol04 - Math 115 Final Exam 2 Page 2 of 13 Name: [6} 1....

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