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07aut-m1sols - Problem 1(10 pts Mark as TRUE FALSE the...

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Unformatted text preview: Problem 1. (10 pts.) Mark as TRUE/ FALSE the following statements. If a statement is false, give a simple example. If a statement is true, give a justification. a) If W and V are linear subspaces of R", then dim V + dim W 5 n. TRUE (:0de '05 W =- V 2. Q“ Mal .1;an A;,_,_~\,J+climv=2h 7h b) If A is a it X 71 matrix, then dim N(A) g k. TRUE FALSE Chum. NCA) 3 it 04 Cab/gmlfl-l 10/0 rig/0% Se lea/ice A. a E" 6.1“ p‘: I!“ L. J U, I J Fr“ . . 'OLJ Maud 0L5 50m. mil lam. k: c) If 3 7E E and V = Span (3), then dimV m 1. - FALSE E? L) M44 out?! luau-1‘4 WOLW‘ (:1) Let A be a 2 x 3 matrix. Then difn N (A) 2 1. FALSE Alt-M. N (A) '5' “1* 91: (LOLA WHJ LI/o {MED-(- We! ‘l’Lt/br—e are, 3 cob/HM”; will.“ 094 («t-Lou" 2. piuD‘ES ( in 0% {'OH 31-le MOM! L36; OU" Mm; owe, le4'). Problem 2. Show that if {5,13} is a basis of a subspace S, then {5+E, 3 — 1—15} is also a basis of S. a -7 -a —) we had {.0 shoe: 1L Ufw V-LJ we flmau’é I I‘Mmoeewtl. 2. S: SPOMA L;)f:3‘ CI = C1 = O I (JP-gfi' MA 5;) __ I; M .«clepeaolehi 7 .3 .5 Me)“ (4.10.. 4kg fiat/{- ~LIILG,+ Spam L Vt All) 3 Spam 2‘2 gar $=aua kW COWEWWOM 9&- 3: a: ' k .I '3 “'3 .53 —) —) 4) —; .4 _; '\ I "v? , SWW k M; : Smev-wi my wists» -) '7 *7 __ __ __ 4L Spay, (VH3 V) 61.4 \J—L: = Zvj- (Uni-‘3) 2'( ‘- ‘F, a} "I’ —‘I 51 Spam J1”! j [3) 0-_/> L4: VTW)‘ V ‘. II-u- a) "" a) I) ’1 -"} Span ( t 0V) VH-o = v t-LJ Problem 3. (10 pts.) Let Z = [1, —-1, 1, —1]T and £3 = [0,3,3, 11?”. a) Find the cosine of the angle between the vectors 1—1: and {I}. -‘2 Cos oL: ”3°.W. \nTE’l-[Lfil --..3 *3 M°w= —l 1 l4: 2:1, (3|:Q+qu =lq -.. .51 Si)- +209“ .3 C C} -. LZF the“ b] skew/U Llama Lu. eta-r 2 - “‘0 1 -l0 2 *lo 2‘ |"3l4‘ ~§03)G \) I01 2 7 !‘3li’w low-2 2~1w o.~2—~é! Problem 4. (10 pts.) Let A and B be some matrices. a) The spaces N (A) and N (B) are linear subspaces of the same R“ if (choose one that applies): 0 The number of rows of A equals to the number of rows of B. o The number of columns of A equals to the number of rows of B . I The number of columns of A equals to the number of columns of B. I The number of rows of A equals to the number of columns of B. ‘0) Assuming that N (A) and N (B) are linear subspaces of the same 1R”, determine if the space V = N (A) O N (B) is a linear subspace of R“. If it is, show that three subspace properties are satisfied. If it is not, show by example that one of the properties fails. Note: You may use the fact that the space V x {5—5 |AE = 6 and B35 2 a}. -) ._7 1)59V om A-an mar (55:5. 2) Sfc—V =? C'X 9 V —‘l a —> 3.) :2 .9 LT; T3} gyros @- \J 1‘ n (a ‘v d -—> C) Let's take A 2 [1,0] and B = [0,1]. Determine'if the union N(A) U N(B) is a linear subspace of R2. Note: You may use the fact that the union is the space {E |A3§ 2 6 or B 35 = 0}. NHL): {x x=0 } NLPJ): {2| Mo} we) NU?) 50 MA) 0 MB) is hole as. amour" 5WL space“ Problem 5. Let U = [1,0, —1]T. a) Find a. basis for a linear subspace V m {E E 1R3| s—c' - 717 = 0}. \!= N05?) £3:- K—- @to'_ 4144025 WW+G\K 1’10““) One, Fibre" | .50 K3- vaficxmefis Own 0‘ 1" O "' 'N NLA‘) :- SP OLVI ( [f I O 0 I J i A; x X.L K3 b) Find a matrix A such that N(A) 2 Spam (3). l x3 are. \Cree I"? J l 0] W12? [:9] #:23er h n n., 49f Med +WW‘ 44.”; load; 3% c) Find a matrix A such that C(A) = Span (:17). I l A=0 “I Problem 6. For a given matrix )9 1#2_1_2 —3 . @n20 _. 3 —6 H2 —4 —5 . _ 0 0 CD Am 3 —6 M4 -—8 _13 w1th rref(A)— O 0 0 2 —4 —1 —1 ——2 0 U 0 (you don’t have to verify that TT€f(A) is equal to the above matrix.) a)find a basis of N (A) "Effie: XL l K;- _ 2 __ _l_ bound 94, NCA) 3 g 0 -.. “Li 0 i ‘- " .9. 0 l b) Given that A [1, 1, 0, 0, of” = [—1, —3, —3, “2]?" find all solutions to A? = [—1,—3, —3,—2]T {a g“; 3‘ 2f} 7 :7 '3 =7} '—.h} ' I : g ‘-' __ t \ Q x~xp+xh« 0+50+%-L¢ O 0 O 0 o l om ClfindabasisofC(A). = Cobamuxvckorredyomie'mg 4-2: goluw“; Law Ply-ol- “4 “‘34- (A) I ~--—\ -1 3 we. -9 i l l"“( I. *8 Problem 7. (10 pts.) Give examples of matrices with the following properties, or give a Short explanation of Why it is impossible: (a) A 3 X 6 matrix A with rank(A)=nullity(A)=3. \00 coo QLOQOO Looiesofi} 10 (b) A 6 x 3 matrix A with rank(A)=nu11ity(A)=3. mafia) :3 1m we ‘W “re—L (A) (c) A 1 X 2 matrix A with N(A) :2 {a} . ‘. OIL N (Ah §5§ => “new: m m col/um 9’1 "44 ”"LL‘W’LF’ ,2 09“.;(1/‘19‘ Q}- Maui— 0144.; 10qu 2.11014 robs Mag] p‘wfl- so -. MA) = 2 - 1:; (”Laws mun ps‘uof 7/ 2812/ (d) A 2 x 1 matrix A with NM) = {6}. H 11 Problem 8. (10 pts.) Let IT) and 13,175, . . . , 037 be some vectors in IR” such that 133 does —-v not belong to Span (€141,553, . . . ,vg7). a) Is it possible that n = 70? Explain your answer. Does the answer change if the vectors "01,122, . . . ,vg7 are linearly independent? --3 --> ._, Ye"; H. Com/[~21 \oe. ‘LLLO-c'l’ U1 ‘— U‘z ? -- a U9? _mmoi LL19,“ 1 ._ "1? [4'5 eat) £0r In: n04- ~Lc> Eater-U 4-0 Spat». (U1) 0n the other [wound '13le are. Hindepemdem‘l} WI‘ bobcat- :O figmea‘r-b L‘Adepe‘ncdemi- vac/140m Lu “230 So 14‘ J n 9 4‘ POSJII Hg b) Let A = a 77; 0—2;? Under the same assumption that ’13 does not belong to Span (51,133, . ,vgy), write all solutions to the equation: AE=E i =‘ A If “'7 _\ A : Fume w s are L - ”Cr-fl ... LLL) \ 4- ’40»: "0 Job-Ao‘u‘ +Lxen ”HM; 0.190%?- ‘aj "‘ ‘a’ ‘0‘" 7.: —-"‘,_._._——"'—':-"—- 12 —,—¢—p Problem 9. (10 pts.) Let 01,712,?)3 be some vectors, and let Let, UFO NO— 13 Problem 10. (10 pts.) Let £2 : [1,1,2,3]T and 6’ = [1, —1,7, 117". Let T : R4 —) R2 be given by the formula: 21) Show that T is linear. b) Find the matrix of T. — t _ “1 1 23 "—“fl ”-3 V ‘j _ A: [leiflelt eSkTeQJ—J l 1‘] 14 ...
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