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201f02mt2b[1]

# 201f02mt2b[1] - MATH 201 SECOND MIDTERM EXAM 11:00-12:00...

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Unformatted text preview: MATH 201 SECOND MIDTERM EXAM December 21, 2002, 11:00-12:00 best ﬁts the four pomts (t,b)=(0,0), (1,1), (1,3) and (2 2) Q C 2 o ‘ l O O C [j [ ‘ ' I l l ,_ D 7‘ 3 I t: =l , 2’— l C+b+ - R Li” A" I l l J E 2 C +13 + £7 ‘ 3 ~ 2 I Z 4‘ (+7.0 +4-€ — a ‘ {41% l A :4, w/xacA ’5 IRON ‘ LL. “.1 72% "ﬂ X W at" 740 /oo[ 74; P74; /e\l:74 Jim/42rd sci/«242m dvAd {OlVl ﬁl- gawk/K: 4. 4 6 5 Q 4 6 I O E :: g ‘ ’ 6 {o l E 5 (b) ( Fill in the blanks) In solving this problem you are projecting the vector: A \ onto the subspace spanned by: 'H‘ e W- -PAGE 2- NAME; _________________ __ 1 1 11. Let A: 2 —1 . —2 4 a Find orthonormal vectors e , e and e so that e and e formabasis , 1 2 3 1 2 p for the column space of A. (b) Find the projection matrix P which projects onto the left nullspace of A. ‘ l (09% a: Z 4: F“ el : 0L :L 2 ’2 I 4 ’ Hall 3 -Z 83 “MIL 4'2 "5 m 4144 roux/L?”ch M017),- A9: [a f, ’§][3‘;]=[3];> —PAGE 3- ~ N AMEz--- ................. -- III. (a) Let P2 be the vector space of polynomials of degree less than or equal to-Z. \ 13(1) Suppose L: P2 —) R3 is the linear transformation deﬁned by L [p(t)] = [ p(0) ], p(—1) p(t) 6 P2. Find the matrix representation of L relative to the standard bases 0sz dR'_f ﬂy I! m ﬂunk“! 4w} r/[Z Ff“ ‘02” N73: 5 T «Jpn! 6,21] 00] / ez‘=(_-o f 07, 9313:09‘1 44:0 vfﬂ? ‘ 3 Q|+€Z+63/ l J 7‘s! 7‘, P \A ‘5‘ (b) Let P be an n x n matrix satisfying P2 = P and 7t #1 be real. Prove that the matrix: 1— m is invertible and (I—xp)“ =I+l—):XP . 5““"’“~ ﬁe “RV—3w r; ,I—AE -. (I-Aflxzo :9 kfyzx. ﬁrm f: Aﬁzyzfy : AEX "'5 (A‘l):E‘><’-:O =5 fxzo é X=O. TAXJ ’MMHJ M ﬂ; rxvxl/J/xu 1‘: fox/13.1 .* (El/1;.“ Mf3-(\f)=O; 3—»? h 0F dob n<=> (1-;p)“ and: A (LT—AB) (I— we)": CI~\E) [3: + 731’] = l—kf—jf. EL+AEL21+>E(:L 474—] [—A l"/\ lﬂk 71(34qu 1 ‘} A >\ ‘ E H - PAGE 4 - NAME: __________________ -_ 1 O 1 0 0 1 O l _ IV. (a) Let A = 1 0 1 O . How many of the 24 terms in det A are nonzero? 0 —1 O 1 Justify your answer and ﬁnd det A . TACK! df‘L 4 “gaze/‘0 7l€rms I; M TAe—VQ are ‘1“ a” 0'22. 0‘ (0/ 2 ~‘ 33 454 '— au 3'24 5733 0‘41 2 —( " (11‘; all?— 43': a44=_l "i‘ dig “24 d3{ alt—7.“ —’ a MA: “*4. (b) Prove that if B is an n x n matrix of rank n, then the adjugate matrix Bcof must also have rank n. 5%=&afg)l / amaw’ é 444%“) (M3354 4:0 % EM éﬁis haan in. ...
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201f02mt2b[1] - MATH 201 SECOND MIDTERM EXAM 11:00-12:00...

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