1998-99 Fall MT2 - MATH 260 Midterm 2 Q4 J TOTAL Duration...

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Unformatted text preview: MATH 260 - Midterm 2 December 19, 1998 Q4 J TOTAL Duration : 90 mins. Name : Section : Student No : 1. (25 pts.) Let W : $pan{(1.l}, l),(l,l,0).(3.l,2)}. '1) Construct a. basis 5 contained in “1,0,l),(1:1=fl).(3,l,2)] for W. ii) Extend B to a basis for R”. 2. (25 pts.) Consider the vector space F; = {c2z2+c1z +30 : shame e R} of polynomials of degree 5 2 with. real cueffidents. i) Determine all values of a such that B : {L24- az,z:2 + 7: +a] is a bwis for P2. ii) Write the basis in pair: (i) for a = 2. Then find the polynomial fix) which has coordinate 1 vector 1 in this basis. 1 3. (25 ms.) Consider the “mm W : .S'pnn{(l, l,1).(1.fl,2j} cm". i') Finn' an orthonormal basis fur W. _\ :i} Find :11? orthogonm projchiun of (1.0:01 on W. < V. 1V.> X. 1' (V1, IV; >¥1:(Vlfl :(vl < v‘ M) x,+ c: WM)"; g main)..- +Pfla, 0:1) J=J (99,031,119 1'3: ‘3 Maui”) wrzflflfl} mud/{H19 < w, Ian-u) Wt: (L11)- 3 __u,n‘ WL= (.IIDIL'I " '/ 3 : (1,0,1)- .— L ('Dl'l.) _ 0,9,; (5 _l ) ; ("El—"2°; 1" E) \Il‘l v < “3.9 ~LJI L‘ t)\ I (”,35r2'5g' o <\_o’__1,'_(_1(lf"”>=3 < 0,3, _L,., _ L} inflow nJo)_ Latina) K [111' ”>10 '._q- -L—Hzo 5; 'u I K Lb ;><V~|V. (”1““)th _ _ 4. (25 pts)‘: Cansidér. thei‘ulnei pxoduct space Rzxz of 2 x 2 real matrjzes where . [Z a] [Z :,])=un’+bb'+cc’+dd'. )DeterminL :L I: such thal [he set 1 1 ' a 1 2 b . 7 :—{[u 0: [1 0],[4 1]}Isanurlhogonalsct. 1 1 i1) Determine tha shortest distance from .-\ = l l I to W : SpanfiS) (he. Find the minimum value of HA — B” for B E W.) az—J ;l3 a+f+or0:0 “21;? :__1 J 2+L+0+0= *3 2 2+ 2A*L_~1 ZQ+L+9+O;O Mi il-[u :]'[:1l[“ “Dz/affi— ...
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