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Test1(KEY) - 2?“ TEST 1(Math 250 A(a Show that the...

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Unformatted text preview: 2?“? TEST 1 (Math 250 A) (a) Show that the midpoint of P1 = (x1, 3’1) and P2 = (x2, y,) is given by: (20 pts) P : P1+P2 21:2: M 2 ’ 2 gay (Hint: Consider any of the tiiangles generated by the position vectors OB ,OP2 and 0PM) m; s”) ”.9 3,33% ng‘,,ovz Me: 0% PM Pu peg, New at: 2’3 , P; M {3,4 , Wmt ta ta? 0PM :{M‘If} We ”4: "I’m ‘ 5,2 W6? MOPVQ R “PM 7: Lag: ) bfg. PM 35 Th? Magnii‘PQP-wvi" feet) "“3 PM} My 9 ”3 PPPOPM: oP+PP 5§>té§m>§ : OPPWJQP’W}?,} yawn) ‘ t - , ‘- ‘i" L(XP§’§(<C‘P \x J M XWX; new!“ P 2 PM,» iii/2.) 16?, {W P “W {I 2, ' ”Em/1 ’ /7 (b) Use the above formula to find the midpoint of f; = (1,—2) and P2 = (—3 ,4) . PafngpPPP :33? , 2. 2. (a) Find the equation of the line L through the points P1 = (1,0 ,—1) and P2 = (2,3 ,—1) . Does the point Q : (0,1,—1) lie onL ? (20 pts) P m?) we» .1, a :gg/ (Vi: Rig; 2:: DP; ”Gt" :(’ ,3, CD} is flnwaaiieé ”in L {tiff y! “g“ ‘1 w (a «t: 3 i1 ’,, m __ p s @egfifl ma: 9M1“: -—> P P; fig, “3 Q «f: 54% LitfiiQ~ 1:535: w, ““353 J "’ ’ /, Pi:—§ (b) Find the equation of the p1ane M that contains the point P: (1, —1,2) and the fine L: x=t,y=t+1,z=—3+2t. PK” MP gig: fix {I 3 w '3“: Ck Q Vi E” i g gag W “it: is} ‘”* .fi fl 3: [O in i f - ‘5 it “/y A; “WWW"? A m (zit/if {ii/f: PM a i {iv 1: i) R) V i: ,5 3 “”5" “Q 3: (i 1 7i; “’5 ,3 N ”' v“? M ””3 m e“? t” . e L, Q}? ”Q? 3: (i/ ”’ijg') M} £3 “E 63% ___‘ __,. V‘ WM: a»? W P . 3 33> PP, :-: a 2 5 e , '3” i 3/ 3W G P 5:; (13%,”"t 1.1 I (3,: 332,} g § § 2 3. The set RV = {x 6 IR 1 x 2 O} , equipped With the following addition and scalar multiplication: (20 pts) _ x + y 2 xy 2, ~ x = )c’1 becomes a vector space. In this vector space, ShOW that: (a)1+2=2 (b)0'2=l la” 21.; $352; a? ,2 C922 :1 .29 t l 4. Let W = {p(x) 6 P2 [x] 1 p(3) = 0} . Show that Wis a subspace of Pix] . (10 pts) (Note: Wis the set of all quadratic polynomials that have 3 as a root) Let 533m ataxia} 6- M] 3:12? {55(43): t) SW35 $1353} 3,; ti} 3 Mm; Lf’rtfilltffil :3 ”’3’? al- iig {El 7: O + Q Q :3 [p M} fix} 6 W Eulagg’. // 5. Let W = {A e M m (R) | det(A) = O} . Is Wa subspace of M M (R) ? (10 pts) If your answer is no, provide a counterexample. (Note: Wis the set of all non-invertible matrices) 5%} g5; mflgfi m Wawfigww: of ”fiaxafll‘i} ) EECr 3; mod” QEQMME “”05““ “5”- Imdaad; tag-fl A; 3(; g) )A,& :t (i; g?) Q W p is}; G; \A/ (has: d.c.'§"(i3uvh%}:§ $43,} “Mt; Attfila:( // 6. Show that the vectors x2 +x3 ,x,2x2 +1,3 span the vector space P3[x] . (10 pts) ml ’Agfifl at}; a Agar: tr A§51%3+l) t— Llar"'i%:3 :: (singlet 53x24» ax 5r «<5 A ,3 a, 3 it. w l ,3 5 a, . ~X 33> Am 51% {at azfilgjx 5+ air a» SAM-am 345115553 ‘5" W "'5 C555 55 iii? 3 AUEZJQBJ‘EQ $33?“ fl 3:5; , ‘ L 3% tgwl ’ a} A; :2 ME 39553 ‘3‘ 1‘35 ’33ng 9W“; Wetter” M Egg—“(:52 3 .rtlzca :3" $ l$ a thomk 0t” 1 1 1 0 Ma vervlfiw Kan—x3; ><J 7. Show that the vectors A1 = 0 0 ,A2 = O 1 are linearly independentin M M (R) map; and *5" ”‘3‘ W‘ ~ M Q :32 3. L9” ”MM “5” «535% r" l a a} flow) //. A 9 3 "’V w as; E if} x; .m A g 5;; E :3 gut/0o) “in “mile? tg5”l:eb<3,i ”3 fair “i $l2 :5“) :5, £533 £3 A %!+:§z :3 O \ .Q—/: i {5;} 9:2,. ‘2 {.35 a; J (A! W Q j 322:} T)” m: A?“ 7:: Q a r 3%; j a 3‘ /// ...
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