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Unformatted text preview: MATH 260  BASIC LINEAR ALGEBRA SECOND MIDTERM FAMILY NAME OTHER NAMES GRADE SM: 11th May 2004. Duration : 70 minutes. Three questions. Points awarded : 20 , 25 , 55 . 1. For which values of 11,?) is it possible to express 4a 4
(1—3 , —b 6R3
(1+2 2b+1 as a linear combination of
4 U
U , —1 E R3 ?
1 2 ’ ( Check your results carefully. There will be no partial credit ! ) ‘ra ‘* °
“5 “m h hrl'H'cn as a linear com lfinai'l'on‘ x p 4. F _1
' 2 4+1 lid 94‘ (13 :,F locate. a 2; H n+7. = 0H2? 1'5 a linear combinai'l’m In? line a¥aremcniionecl 2H1 { all values on} 5 Since l 2. Consider a vector space 51] with an inner product (0,.) , the norm  :1: H for
any 3: E 93 being deﬁned as usual by ] :1: ll: (3:, cc)”2 . Given a, ’v, w E EU, compute
 2’“ + “U — 3w H if H “u = “U H=I w ll: 5 and (um) = 5a (Elﬁn) : —5, (115,119) : 0. ( Check your results carefully. There will be little partial credit. if any at all I ) 'l
”2H+V3W” = ('2“+v,%w/ 2u+v—SW>
a1 ‘1
= 4uu+ Hvl1+ 91wll+ 4<«,v>12<«,w>#€<%~>
= 4.25—+25+9.25+4.512.0 “6(4) =' 14.25+4ol5 = 1645 "AMI”M“ = 4.5 : .20 3. Let Vn be the vector space of polynornials of degree at most n with coefﬁcients inR. (i) Prove that Uri : {106) E V71. I100] :p(_t)}
is a vector subspace of V”. (ii) W'rite down a basis for U”. (iii) Let the inner product of p, q E Vn be deﬁned by m = [1 p(t)q(t)dt Find r(t) E U4 such that the set {1,1~ 3am} constitutes an orthogonal basis for U»; with respect to the given inner product. {l} [or ahj I’IQVL ' lune)
Consequeniij 1’1 _
.a a, fuel/l" and aﬂguq :{h zapff? WL AP{*)+ flat‘1") : Q‘PH)+}“H') : A“) _
“WNW Mk,Thi5 being in” for arbii’r’ar‘tj 3'2" (HR and F1 ‘1‘ un We conclude “11+ H“ [5 a Subspace.
(ii) pm = ~,+a.t+ «1,121+ e M“ if?
PM) = “r“ + “354,9“ = a” «Anni 439+ _. for)
Qinian‘ﬂ‘H
3’ “4% + 43i3+ asylum... : 0 f”. ‘H J, 5 {R‘
We conclude Hut PH) 9. Mn {H H is o[ ihc form (iii) If hoh'tr. ‘Hn‘l’ Tukl'rlj a: 9 J W" ”Ham V'ff) :: 330t1‘i' gjtv H... 1J.r'——> 2 ‘r
a
0+ «2+ 4» qqp’c «I {12151} ”a“ } kit} i, was checkul our 1 J. 191:1. P 1 a
a+§+§~=o W1it*3+}"+ ...
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