06spr-m1sols

06spr-m1sols - a saw-rmnmmmmmwrr:-:-"" A...

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Unformatted text preview: a saw-rmnmmmmmwrr:-:-"" A A ~ ~ 3 1. (15 points) Suppose that —3 1 0 “2 ‘3 4 A204M11v=1,w: 2,X=_2. 5 - 0 Calculate each of the following expressions, or state that it is undefined. (a)(Av).w A?) e, “27'— A [mg 2 mwgr - \odsc Q 6:. [[23, So +$Gs is SMCQ UMGOE’Q—Em 6:50, (at HQQAEJPV\I 563 +663 . £3 a SCQ‘QF‘ bug? :5 Q geeks rj 3 (C)Av—3x .> ,_ (7'3\(—73+ (choc: : 4‘5 * ‘1 "- 0> (so Jr dot-A»: —() 2. (10 points) Give a. complete definition of each of the following terms including the correct context). 5 (a) Null space. The no\\ cc I A 13 We g6“? OK“ \JQC‘i—GFS a?“ such W054— : (EQ— QuA MNA Matt??? 5 (b) Dimension. The 0Q:MQV\§€GV\ 0C Q Subg€cxce V is We (ADM‘OGV @Q QQQScot—s M Q (00535 ‘Qaer WQ MGc/Kfi‘t‘cjm 09 U {A641— wQ Wot-MW \lQC‘ESE‘S {A EUR—g Act/1M“? Mod waSQQC‘E CGVVQ'GEV‘S 3‘" ‘w‘te‘y Maw, wankers ; —- v-w—- " a *:- A' . 1 2 —1 2 3. (30 points; 3 pages) Let A = 2 4 —1 1 . ——2 —4 0 2 I O (a) Find rref(A) and determine necessary and sufficient conditions for a vector b 6 1R3 _ to be in C(A). lZ“le[ \‘Z-l25:‘*@2 2 q i +91 4%..» 00 l “was ’1 "*1 c5 72. b3 #2?“ O o. “2 G \93-2‘219M2Q2 K 2 0 fit “‘51!ka- _,/‘3 O O ( “"3 “15‘ “1‘01. (3 o C) o —1EJ\+2lol-k-\os l 72. O *4 i. a .— mcm w go :33 bk (1)) Find a. basis for C(A). ' FREQ/O“ \nas pivoi-s in Coleman; 1& Gmoifijg. 50 Que {06565 Car camsfsk ac -- Column? at M 3% 09 A ' Li (C) Find a basis for N(A). Ta ‘(AONe : 6 we neecfi Xi+2XZ'“Xt—{ :0 , ' ‘ ( SQ X’i XL )2: 7K1[(;]+ KL; [3 3‘94 a O . so we {mug Cm— N(A\ is (d) What are rank(A) and nullity(A)? What do these tell you about existence and uniqueness of solutions of a linear system Ax = b? ("amicCAA‘i don-(604“ Sic/ice - C00 I: Q Z-‘ciEMQVKSEG/EQQ guloSEDGCQ G'C' May or May V163?L €XE$+ (fleQmofifuj cm Mew cm 2 deM(Nw\ :2. we no“ gfmce Wefiv‘cere [4‘33 Maw/(Y Vader); M H, Sob‘i‘fomg awe mGUGt- UM.ka (e) Calculate - ' v " mar worm +24 2 Li -—I l S: “I: 7—“ +qv6—(fl +[.‘( ~14 ~Ll=0 song-'24 (f) Using your answers to previous parts of this problem, give a parametric represen— tation of the set of solutions of the linear system $1 + 25132 — 333 + 2:174 = 1 2331 + 4232 — 333 + 334 z 1 _2$1 _ 4552 + 2334 : 0 ‘ t ‘r-b a K k a r ‘ BY 6)) Xv: [a] '15 q Par-traders SJo‘th/zk' 30' “$3163 (A $69 Qu SOWHSMS Can U C I k. - "2' ‘09 mrctten {Falls—S C; +15 5%) ‘1, sffiéffij> 1 O I M+€‘ we 39% 6Q Sdbfigms- meows chokes Car -~I- u I $6 OA(Y Involve — K3 , 15-: “9% ’4 +3 I \l G QtGFS wt ___ C 0M ear/1 @n h .-:-.-.:::mafisnmvnnmmmlmm:7'2""""2""1"""- -' " '- ' ' 4. ( 10 points) Find the matrix A for the linear transformation T : R2 —> R2 which satisfies T(E])= [‘3] WED: [5]- ‘ffir 0mg Mowgcfi sob‘Homf See We Pmcfi‘ce' QKQcM So[o{’EGL/\S" Rea-9’s QMQ‘E’flqu—i 61 t0 ' 71 ,_ Via +‘o :_ - lads : [‘3 AX[ 1] f '26 £01 3 . :7 160} r S.- a \ [cifléflifiig 1:] Li (ERG/l: 0 MQfl+ +0 SQ‘UQ G. \Eme Cler- SY Lt UCN‘ECLEIQS q logic? \ O <15 ‘1' \ “3; o (5 0 1’- ‘ 5' amner 2 i <3 0 L1 ~22! a z 3 O O O 2\ 5’ “223 ~40 K 153 O R o a 2 I O (3 ~39 O O \ 3 O ,BQq C5 =-S‘ 5 4-153 0 o 0‘ 4‘ _ 5. (15 points) If V and W are subspaces of R”, then V +' W is the subset of R” defined by . V+W={V.+w:vEVandwEW}. Show that V + W is also a subspace of IR”. W33 3'3 mi— W SauMe Q8 Vflwfl I 36% ix'camqtms [0,11 ale/(Ci \0‘22 L‘GMmekl ® qwcfl G W since are guhsqtces, h.) - SQ 010*36V'EM‘ Infim‘fimmfimwafi:ujuf:“"'-A-:u-u-. ‘ ., U (2) Le‘r QGEZ and 3'56 Wm i=6; +5.3 go? some “(WWI amt (flew. Se c431: aCmfi : (1'0 e at; New (3:, av W ‘is dcan 0%va Scalqr MuH-{chafiom [qt-(01 _ (:53; EM smiles—Rf. Se C‘i : (Kiwi? 6 \IHU‘. Leér (:65? and 3253 e\]+wm Them :32 {\jfifl‘t _b Q_"'3. a = \j'; +ma gaff 3C3th {7t 7;): @V QWOQ juggle GW‘ SQ .5 _‘:=. Mb \ :2? +35 3: (fi\+U—j\ 3 + (651+fi2‘ : ‘L\lz\ +(W\+wz\_ New “U: +G~L€£U game V S Ufldner' \fecjror CAGhHCM, flwal Ci * it}; 6 W gEMuat-hf. $0 3226’ 4‘: (“\éimfif) saw-$.13 QVWJ“ _ 6. (10 points) Let P be the plane through the points (1,1,0), (1,0,1), and (1,0,—1). Find a normal vector to P and give an equation for P. Hint: recall the definition of the cross product: ’U2’w3 — ’U3'w2 V XW : 113101 —"Ul'lU3 . 111102 — 712101 eéMéaic‘CYMjf Y3K] €ro ‘6' Uge {Ms Gama wéH/mxfl' [M’tewt'fifj Lei/ted— Jaime maQ-Q‘H'Gm LMQQVLS, m VOL; :‘KeG’G‘ +0 Wofi- YOU agar ' 7. (10 points) Circle True or False for each of the following statements. (a / False I T : R2 —+ R5 is a linear transformation, then knowing T(e1) and T(e2) must be enough to find T(v) for any v E R2. ' This f’S eméwgk ImNQGrMaH'on ire @6an fine «Aachen; gov T '3“.- (b) True /' If V is a 4—dimensional subspace of R5, then any set of 4 vectors in V spans V. ' TlACS maulcl [De ‘H‘UQ (by VCOP‘Eg‘Hfin [2.2 0¢4P7€> I 3‘3:- Saaécll “ends/q limeafixf lWC£€!E@aLCfl€r/I{~ Se“?— I Ll GQR‘G‘RS “A \l W. H (Mme/@123 ”- ‘ If A is an m X 71 matrix, then N (A) is a subspace of R”. NUS is a solusfcme a? ll?“ , mot-mm? (d) True / A system of '5 equations in 3 unknowns must have a unique solution. I"? l/rcujca Ck UmEqUE gelo‘lt'i‘w lad? Cllongt‘l- ln‘un ‘l‘oi Q SCUT Egomele: C373? L“ t +‘ m3 ' i as (MW 36 o co (e) False (fig-:78 Y 1% .m Any set of eight vectors in 7 must be linearly dependent. Twé \01 PFQFQQE‘l-iom g-L’l Gm [155 ...
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06spr-m1sols - a saw-rmnmmmmmwrr:-:-&amp;quot;&amp;quot; A...

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