06spr-m1sols

# 06spr-m1sols - a saw-rmnmmmmmwrr"" A A ~ ~ 3 1(15...

This preview shows pages 1–9. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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£€[email protected]ﬂ€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 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 9

06spr-m1sols - a saw-rmnmmmmmwrr"" A A ~ ~ 3 1(15...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online