Midterm 1 - Solutions

# Midterm 1 - Solutions - 1 Let V P2(Q the vector space of...

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Unformatted text preview: 1. Let V : P2(Q), the vector space of polynomials of degree at most 2 with rational coefﬁcients, and let [3 :{1—— X + X2,1+ X2,X — X2}. (a) (6 pts) Prove that B is a basis for V. 4551mm; l i" )i/ l if lair X (T. (2%; X i x/ E ” ,i g mm; {a J a 7,, 6 , l w T x: 1,7— l «A. 1* r” n} is Vii/5' r t- " i if“, lhw éMcc« “ NV 1 w .1“ ' «I l ,3 ._ v a, 57W 44% ail/l W 1:} i5: a; l A I}: gﬂfl/I / (b) (4 pts) Write 3 — 2X + 5X2 as a linear combination of elements of 3- _ ‘ 1: Mia) 7’" 3 mt) Mi)” M [a :1 55’ ﬁr :2 (l “(57) jag 5 J. in é: *7 f :1 Qw- k\\_ 53%“? Ll. lac/i 2/ 711/65 4;. l [ll (dis/65' 1-;3, BLZX +5: A W M + 2(2) (ax/ls;le (fir)? ' 2. (10 pts) Let V : AJQXQOR), the vector space of all 2 X 2 matrices with real entries. Let 7 W1 : {C b) eV mama}, b C Wm get/per}. Then W1 and W2 are subspaces of V. (You don’t need to prove this.) Prove that V 2 W1 EB I/Vg. Lei/1L l} i WM] mi; 4 I Bur?“ :94"): ﬂ’fag WV 0; / ﬂ é/ '4’? Z jig? 0 am] thbtﬁ’ 4/5? 5 yr HUM M g a Thwahﬂa / W, H W, 5 vi? @éI/WZ/ ZﬂvL§Q114/}(jé/ :5”. W, 0W2 film: ﬁrs a?) 61/ W6 WW '7'” “WW Wig (MM/{4’ ED \ / / as a ﬂag we mwr twe ﬂ.i#/§ ﬁ)+(0 é] ﬁr } ‘ f , i 3/ / AX / ’ ’ ‘ ‘ WM? (3? ‘1’ {‘5' ‘ jﬂzwf fhb’: “5 ifi’éﬁz? n? ’li-é‘? 119 I: 14’} 4)” C e?) I . ,_ .: V9; 5:, W 517;; (I; b P 4/ :5“ C/ , Tit/M5 b V v 4 _:@p4 h ad é@w} 5(7 w, :4: (414340} A 6" WV} {Ag/«2‘ : “ng r”; é re ' far 3 I > W a ,«v . r 7% if 5'" .9 f5 (1/ 6 5AM a”! e W; i , «a? is"; ‘5” if? g "t r’ 3. (10 pts) Let V be a vector space, and let W1 and W2 be subSpaces of V. Prove that W1 U W2 is a subspace of V if and only if either 14/1 Q W2 or W2 C_: W1. ( 55% (AA/26> Wi’ (ii/l}; M‘ W, ’4‘ I77” I Wl 9: 1427/6117 W; U / WAR/ﬂ (5’ 4 :éu ? 1 g *A/ﬂl U “if / w mg? 15’ r! {irrfizé‘éf WW; U 5%. 6 M/W/zwg: (if (“v-NW) New“ tamqu r if [5% 4 faﬂﬁ/ﬂém (7)7 H [gi/fi‘ﬁiéz 96 +7 W U . ’52 Meg/1556, {:7 {xi/Ms?!“ P 9 J34, ‘ ‘ , a, if?) r: 'l /ﬂ ’4 l A‘ Hf!" J, m)» .2147" 3'MM {ff/fr: Harm (7‘ Mfr“; 6" 9W “(7 r”? we mm M 5/ vet/*7 é I L I TQM/Ware we mar WW? +7 W ~ 5”" W256 X5 W Aw) we, Wit/é “4% 5 4M; /!/'i/{:§ “’1” ﬂ" / / / / 50 \$715? WI 1 “MM W, Elia 9??" W W; , W; W: ' r H a f , . a La KIM/f Mfg???) 64715354“ W, {3% 4. (10 pts) Let V be a vector space over C of. dimension n. Recall (from a homework exercise) that V is also then automatically a vector space over R. Prove that, as a vector space over R, the dimension of V is 2n. 7 L697” 5,54,, ’ r r/ (£7 . ML l/ 3/ 61%?” f . 2 I (ll/3;!“ 6 )a (4V “7 {Al/y {/{I/ a _ 'f M” M; ( {/Méx WI £3315le 7" fl (mi? V 4 g 4 1/63 /\ ﬁrs/(M igzvggf ' » \ UM Va TIM/l V?” (“I (’{I 7L7 «« a '7! 4/1 MA ﬁ/Ej/m 513/ v, . h) r- @5214}? k/ 1116/ (473 W/ [5k :1 7L- Zﬂk 7gp M €71 vantage Ila» llwgo U“; [4,+Lt,)l1,+ ff” [at] + Lam, m, +/,,+~4n//{A cipbzwjﬂiflzjilwﬂ/ éo véwn (/32). ( Tine/V; m; l/ f/ﬁldfi ), alumna/7 ) M W [Ag/W Maw 41M! {4 “’ Alia [4,3 {it}, g. .1 at" I: 7gp al/m/élla/iéwzu/ é/l Th0] (:2, +301 1:) a, (at; t it) a a Main; K wt 79% gag/1 k/ 7%}; égg,,{::mg /‘ V , , a“, a» Q) MA: W \$6; the 41%: f5 (ii/Y? m. A , \w él/léél 45114., H7 if? /i/1 M ﬁzz/er W, Wit lava O lav 77%, at: + 0 M <3 hmlhwawéwtwtamt Cgéflﬁgﬁﬂ 7i” 631/6; K I Tim; ﬂ :3 {/71 m NW 5*? 7% VI if; £ng 5W2 V tiff ‘ é? Willi“ z? ...
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Midterm 1 - Solutions - 1 Let V P2(Q the vector space of...

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