Spring 2007 done

# Spring 2007 done - Johns Hopkins University Math 201 Spring...

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Unformatted text preview: Johns Hopkins University Math 201, Spring 2007 Name: Section: Midterm Exam 3% 2 Time: 50 minutes No books, notes, Calcuiators. Piease explain egrefuiiy aii steps Eeading to year solutions;3 or risk 3105ng credit. Probiem 1: (6 pointsm2+l+1+2) Consicier the plane E in R3 with equation .3: + “2% +133 ﬁ 0, and let p denote the orthogena] pmiection onto E. I. If (U1,u2) is an Orthonormal basis of E, write the formuia for phi} in terms of U1 and w; {Where v is any vector in R3). 2. Find a. basis of E. x 3. Finci an orthenormal basis of E. 4. Find Ithe matrix for p in the standarci basis of R3. "-3: {‘3‘ List?“ 3 “sh “2 s . I i \ : gig" . :' g g a: (a a e"; i ‘2 2 m ,2 Ag ; M 2: g i 3 W s , k * K @k g} a w , M g y‘éjé’g: g4} ; if: a UEWEEEF’ J£A§m “fg 5"“ am" {3 w *5 {3% WM ‘3 g X r g s m E E m f i 2 Vi? @gmif E if f N £ f g I i I; {32 9&5; “3 \$47k SW W g 3 \ jg ,, I; “13.5 f f g 2 ’3 g1: I! _: ’” r: E "‘7 3i” ‘ 3 E 5 31:: 5—ng 7 v ﬂy"? ’fé M .5} Mg "Uéfgﬁ a : ﬁgs/{3E ' E! If} 32% ’ﬁ E i f r St; jg} £3 :2 i Me, 3% “ﬂaw 5:; .‘ Lﬁm “53%;: a, ' 2,5 1 32 . fij f” x a s- 3 g 5 i n z . g - 5,, i 332%; g f fig? 3‘3? v E 5&3; gamma-gm. 3mg“: ( gm 2% gig 2mm. “mfwf; A frw‘yzg 'W‘M I mg; ~b§§i_‘g ‘ is; rug; M"? 2535? g g‘ is)? 5; g 6’31 gm? 2 x 5 ’5’; g a} 3?» a5; 2 “if: E S a g i E: if; a gig? {I ﬂ 3"“{55 ‘ T; V W“ ' \ ,1"? if? A ,, , 1’ g i g a x 3 g g i E {ﬁfég 3% {jﬁfg gm L?» {@353 5&sz a}: ‘ g i i s f f E «w M ' if \fm m mi 29% mm 1‘ i m :9 fir; E,“ h a: ‘ ,é E‘nggﬁf Etggugfw E Egg fa“; § ng§ '3“ J" §é§ E“ 123‘ a 54-3? 59‘}: M“ i} if g y \ g g; 5‘s 3 s ‘3 §ff 5”; a i 5:” “7'” 5? x W ~ Problem 2: (9 pe§n\$523+2+1+1+2) Consider the linear space P of paiynomials with ma: coef— ﬁciengs. 1, Are {he foliowéng subsets of P iineax‘ subspaces? Expiam Why. 431163 sen E0 0? poiynomia‘is 33 such that pig) 2 O —the set; E; of poiymnﬁa‘is 33 such {hag ME) : 2 «she set P2 of poiynomiais {if degree 2 or 1633 Consioier the linear map f from. P2 to P2 deﬁned by : p”{:r} + 3p’(r). '2. Find the kernei of f. 18- f an isomorphism? 3. Find the matrix for f in the Staadard basis {1, :c, 3.32) of P2. 4. Prove that the vectors p: x 2 + \$5 pg “2 3, p3 m l 21‘ + 3x2 are iineariy independant. 5. Find the matrix for f in the basis (191,392,193). {g {Elm ﬁrm? {EA}; ﬁ% if \$535 A ax.z«§»ésx w: 25"": W" 2;; § E. i f 2‘ gﬁi {i m f 7%- fégﬁﬂ: "E 3&3 iafgiléifggﬁb \$415,) a: 2 9 gm»: . ‘5' = ’ Ms_ E f} w A; x . ¢ W . . [g ‘1‘ f6% «v 3;; fkéf‘ﬁéwgg j; Bﬁﬁw t Jig; “45% {3% “ g: iv??? ’ i J: g g if ’6 ’E n: g” g; is?" :2? f {E i, {afﬁx} f? C} if; N31; :5 x If?“ W5 a \$3 i€§zﬁi:é\$? g m ‘0: a Egg/i I. 1 D a r a r. m 5: w 2:2 at? 3% WW aim Mix A? rm; a H a 2 ; U {L a}; 3 A {‘2 {as i u v a}; ' .w‘ 1 2i ; “ﬂ Haw; géfj’ﬂgﬁi £33330? .; a ‘ ‘v‘ f" ‘ ‘. Cgégngwgaz +€-§{1§%2’£f31i}\$§ 5,? ﬁx an; ‘ w I? - ﬂ gig ﬁat agkgzifis gig-swig}: f3§.}‘:13€-E'CG wﬂ'gﬁi ’3 \y QCE+3££~§CE \$47 {:EfiC’}:<} 5W; 3%? if} 4 ﬁgz‘Ciﬁﬂﬁﬁg .a ﬁmvaﬁéu {ﬁﬁéﬁﬁﬂﬂ‘ﬁ‘ ‘ _ a; . t ‘ 22’ g f” > 7 ' ' i? {53% g5 g 59- } 3)} EM @5277“. U ) if, 9&3 we, mﬁa j {jg W a g M a ' gr \ S f an; ‘ . ’2 3": ; {ﬂit G I} {£53 {33: «+52 .1: mjéj u g? E» f p 5 "2 W Eggmﬁiki gﬁ Amp. {ffgwg’ifg}; Ag {E 3 “ (LN Problem 3: {5 peimsmi~év2+23 Consider the Ina-‘grix: Jts column waters. and iet '3J1;£-’2,’U3 denoie 1‘ Prove that '£J1;E)2,'U3 are Eineariy independant. 2. Perform the Gmm~Schmidt process an {a}! :32, \$23}. 3.». rm v 1 . w m «3% 2;, “Th . a; 9mm w G a 3 gm 2 . E R 5 Q E Q 9... M E}, {Egax t aka. \$5.... 6 2. ma? » . n}. w w §x~.€x:ev W 3%. a. 09. .n, C W p M :2: 3...: . .5: .E?,;; g! iiWMW it?! 4% ‘ €3.35 ﬂﬂmﬁé m a 3 ...s..es§§§1 59:, a w? E? R man” a: w? A Lag-u. gaggigéa ‘ 52% % my W m um a, E izﬁgr; xiiwwﬁ 2E Jw x 4%.: a: 5 m3}. A m an? x 5135:3332. L3 52:3,? 151m rib: 2;.» ﬁgiﬁggﬁﬁnéii § ii ,5 i. £5 “V m fix? :9 WU . {EEK Egg; ,,,,,,,, L x {ii w. EEK}; ? M ﬂ w .3!!! 5;. ﬁg 3.; 5 h El 3.21%.! g A 7.... s”, E J gmgwf R 651E gruiéik AIW Aid nary v. “£5” We at a, , \$1. ray mu 3%; L3,: ,.. E. k «\ 3...» mu m)» ...
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