midterm1sol

# midterm1sol - l.2" it i Nuun(l poinf 22A Midterm 1 1(2i...

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Unformatted text preview: l .2"; -\ it i. Nuun- (l poinf): 22A Midterm 1 1. (2i) points total) a. (l2 points) Find the solution set, of the following; hyﬁh,‘lll of linear equations, and write your nnswvr in vector form. .1: l—‘Zy+£l: : l b. (-1 points) Deﬁne “pzn'ticulnr solution” and "l’ioinogencous solution” to :1 linear system. c. (—1 points) VVl'ite one [NLl'tlLilllill‘ bOlllthl] to the above system, znul one homogeneous solution. 5» I, I L'\ “'3 ‘Lf"; \ ’2'" "‘3 \ {WK » 8 (L14 \ 2" 7‘ “J7 o i w :4, ——-———>; O «x. w" 0 “i ~ 5’ \O o o \ :3 \ \ ’0 «Mill, \ O I,» \ i 4— , q ‘ __/ CD ~\ 0 o o \ L) < O M (1/ , j X . lira/v [ l ’2; w 2.. “‘:\\ } 7g "— "4 \ I m) if 4 “'l'i‘"‘~ 6/ KAY? 4/?6\ *1 ’3 {x \f“ ) imam \ > \ , q. r‘ , ,1. . ~ » ‘ ﬂ \ _, / r\ (if 11WLZ 1/ ' : v- \ / (7,7 ‘ i I ! ~ I; l 7 r P" ‘ ‘ r. v- i. » gala“); ‘0 ‘ ‘ . vA“, ‘t- B 1 l “‘ s l i .— . ; . , :4 ~' ~ Qu/ 3;!” xjm I ‘> i \ " (M) . / V» u Il\ l / ‘2 I / “i r ( ~. \ / ‘ l . q. \ ‘ . n, , \ \ J 2 W Va, i, i il'KI.) 2 K m,‘ L Q Q 2&4 (2m \/ L K ..\ 7’ a 0 0 A1‘/ a {A l.“ 1 U) points tuml) For uzu'l nmtrix in, RREF form w c. (3 points) Ulllpty 5017 -> wt. :1. (3 points) a plzmo. b. (3 points) 2!. paint (k. . 0 ‘ C ‘ I \ I) O \ 11 C) (,3 Li» 3. (5 points mm!) Deﬁne the (mm-hum of 2111 n, :.< 1), matrix 1\[ 2-. ( (If'l‘(.’\[) \$271111) \ 7% “:5 ‘ o E . _ é ‘fu’ '1 j \ O 'C) (2 x U L.) \ l (C. Va“. VVILLt is the anti-trm'e of AI r 1 v .‘) ‘J 5 IO U 8 4 ( w *1 l~'\l l typo of gmmwtric uhjwt, writ“ nu :mguu‘utvd hose rmlutinn wt ("Ul'l'r'hDOIIdS t0 the gvmmtric , 7 Inj) by 4 8 (i ‘) l. (‘27 [mints thrill) [ll-f 1 1 , l ‘2 :5 l 3 u ‘2 .v ~ ~—l. ,l\[ < r 0) “V V; ( 9) . 3 U l) 1 “ l “ Czllmlluto mull uf Hm lbllmviug. if possible (not all will lw possiblv): a. (3 points) The (‘osiuv «)f tlw angle lwtwvcn u zuul v a '3 b. (3 points) .lu w 2v v— " C. (2% points) AIu 2:? d. (13 points) NV .2? c. (3 points) AI‘IN :rs'f f. (3 points) (NMW :2? g. (3 points) N"1 :23? h. (3 points) A vector that; has angle Ir/‘Z with u 7—: I? i. 3 points) Dvﬁuu a linmr trzmsfommtion T : R3 M R2 as T(:1:) a: AU. \Vlmt is 711))? LL *1 . 3 I 1 u ~—-—~—\ mm "“_“ VT,“ L‘LL! (‘x/ ‘ \3 M Hl "\ ‘ \‘l'l \J\L\\ \\ 9“ l T,— I J \\ l\ \ \ L1 1 an “‘1 K H \ l. . , 1 L t. _ ‘ I) x x I “I ‘g (j, 3. it“ a. v], ‘ ; g 1 : ("AF {3 (.2857. ) 5.: l ’3 l r , l a. , 2 /' x 1, WE V \ 1' K ’l'ﬁ‘ﬂ.“ ’ w w -* 2‘\ 'l r\\ l ; \ ‘- l l l L) l: / l \ ~ ~ 1. / ‘., V \ m :4; . A ,7 \ l ,l 1“ \V‘1 /» K 4'. \\ \ My”. n L' It \ \ .. -. I I, 7 M ’ll‘. ‘ r"/\ ‘l‘ f\ , I 3“ #l\\ ' l Ml ‘ l ‘1 J ‘ \ ‘ \ J \ l I \ l l l Extra wm'k thH‘U. F t), (1:) points mm!) a. (n points) Lvt L lw a1 i'lltivt’iuu from V m H'. \tht «luvs it 11mm for \ L m lw liumr‘.’ l). (10 points) ()nu nf Hm. thilowing functions is Iiumr, and thv ()tiu'r is not. For Hm one. that is not lim-m', prove it, is [mt Iiumr. and for the 0110 that is lim‘au', pmw it is limnu‘. .1'1 ‘ fl' - .1?- M In ) ‘7 < l 3) I 11:3 1. L ; R” —+ 5R3. 1.; ’1 + I -v w m» _l.' .". 11. L: 11¢." ~=~ A"), Li 1: )< 1 3) .173 ":1 m x», A i, a v \ i’ " ' ‘1 4 PK i‘ ‘ r‘ ‘ ‘ Y ' ’ \ O\\ (x \. iii‘Q» .i (My 2 "“"K HZ. If I i ' ff) r t z, ', ,1” 7-: "mi , ‘ \y‘. )1 I \J iv, . _ i H ‘ \A L" ail/C. i (JVI\ ‘ i L-(.,L’~\ \ «JL( V3 [ i r i y mi ‘1 i n") [T . 5‘ . iv ;; w ‘ c (\o‘ ‘ 54w. .1 x g 0‘ CW» ifkw)‘ ‘ I ‘ x 'i r) \ 2"\ ("X a; V: {R \) ' WA» L) I, V (I) ; \”\\ i3>\\ ’i",'\ i t "'2 H! r) ‘l\ \ v_ LL‘ {r is ii; \ . . \ J ' d ‘ ‘ \ 1L \\ \2 , 0’ ) i U. L.) I D ‘J m ,1, e' . {AW‘1 u) . z “W 1 "~ U . ‘3 t A II 1} i f ‘— i-Q.‘ () ) {I \{ ” V\ 1’ / ‘ \ ,v ‘ ‘ x I“ “ i,‘ _{ i \r t g m f /,. F ‘ ‘ w M M, )x/ \ {\V k 5/” J ﬂ / i . ~ « x i . . § i \ ‘\ i J 5 ,‘ ,; / \ K i I \ , I \ ( i‘ i \I,\ I, 1 ,w- ‘_ " J — . (i. (‘1‘) points toful) Lvt u r: and v t ()1). a. (7 points) Find a and I) suvli that (I, ~ 11 +1)-V 2: < 21 lim'zn' systvm of munitions to solve fur a and b.) ’7 g ‘>. (hint: svt up b. (5 points) If/l' : R2 m» R” is 2L linear n’niisfornmtiun such that, ’l‘(u) 3 5 7 1 and '1'(v) :2 U , tﬁimi what is [((q> )'? 2 3 ‘ (PX. ‘J‘w L r)" i "‘ JR, . f1. / , ("J 1 x3 1 x6 /’ 2) K ' 7 “1/3 it. - 0 f i f.) “:3 O a \ ~~ 7,, i. /7 \ ‘ ("ﬂ / ’J,, H "L a “M \ \\ - y y A _ ,7 \ r01",,/ «1“ _ - K «3 ) i " ‘ “3% "AV ) ~ 3‘ (my) " ‘ (WV) ._ "1 "7 f ‘ 4 3‘ '1'7\\ ~- I \ a i 5,) i (r "I . I? II q ‘8": ‘ ‘ ‘W :1 f? i “ i if! ‘ 2 » g\ L, , \ a 1, (if? puiut’s l'df.ii)i'1w Milnwing svt. with rho addition and N'éiidl' multiplimw Hun dviim‘d, is Hut :x \‘Ui‘l‘m‘ Spilt‘t‘ ()wr 1R. Provo that it does satisfy flux ﬁrst? (Wu aiximns of 21 \M'hn' :ipm'v givvu below, but (luvs nut satisfy the third (you dnu‘t uwd f()(‘i1(‘tii\' any of the utilm’ sown). V w { :i-tuplrs of iutcgors } J'i ll: ‘1?! H11 1'1 (2171 .173 + [/3 1'3 + f/g , ( 13 ('1'3 J':s 11:; l':x Jr 1/.) J's "H, a. ((5 points) Show (prove) V duos Satisfy: ( H) (Additive Closure) u + n C: V. (+v) (Additivv [llV(‘l'Sé‘) For ovvi'y (L 6 11+ m -: ()V. U there exists m G V such that b. (3 points) Show (provv) V (1005 not satisfy: (‘ i) (Muitipiimtiw‘, Closure) <:'- I) E V. \ ‘ y ) f , ’\ I v H '&1 v LL.» J‘ \\ j 1 [HF ‘ gzhﬂw {[0 [Ex/(3 ,i A x \ \J . .. u." A [\‘1 i \5/ L ) ,r'. 1,) ) l‘v’ \3 I \ - “ i, m “ v “’7 “i \ ~~ \ y, )5“ ’4 “— F '1 . its,» m.) -- 2,141 3 «1:; M ‘ cont-Minu— I ' u A ~ "'5 ' I... {,1 . a , ‘1 v . SML2/ M + M) " L) k u“ w ‘ F (a ’ “R K; H‘ «E i‘ , {i‘f— Mun?" L." \ we; L' ' R. (2 points mml) Wizilw studying for This hast. was tiu‘z'o some tnpiv you didn‘t HHdvaud. imr rim; [ind .m "ii-11.1!" mouer xmd it nva Him-1’ \\'11.1r was Tint riiiug'.’ \Vimr ll‘uHil‘ you Iiiidi-i'htmd if? i‘iu‘n‘ is LU wrung {imum‘ m riiis (Jilv‘hfiﬂll. .md Iimt; ilU‘thiUh hinnk amwvrs. . ,l“ W i T:‘ 1 i L; in 1. .1 YK («M i "y ...
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midterm1sol - l.2" it i Nuun(l poinf 22A Midterm 1 1(2i...

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