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Unformatted text preview: UCLA Math 33A Lee 2, \Winter 2008 Midterm 2 Feb 29, 2008 Last name: First and middies Names:
UCLA ID number: Section (circle one): 2A WONG, WANSHUN Tuesday
2B WONG, WANSHUN Thurséay
2C KWON, SOONSIK Tuesciay
2E) KWON, SGONSIK Thursday Instructions: 0 Fiﬁ in your name and eircie your section above.
C You are aléowed to use your brain7 pencil/pen, eraser only. Calculators are not aélowed‘
c ‘Wz‘ite your soéutious on the midterm papers as possible as you can. 9 Show all the necessary steps invoived in ﬁnding your soiutions, exeept True—False questions. Problem Points score
1 10
2 a 7
3 i 8
4 7
'5 8
Tom} 40 I Problem 1: {10 points, 2 points each) Indicate whether each statemth is true or false (you need
not justify your answer here). 1. R2 is a subspace of R3. g l; x a _ i f “2 3 . i ‘? ’a < “1' , ‘2 ix: x ,w ' iii a e g éiiﬁr ’ a m a? r3"? m: e, a ﬁe“ er a; a
g . V  f i
2. If A iaawmetric (Le. AT m A) and invertible matrix, then A” must be symmetric as well, g g E _ Hi , ﬁé’wﬂi ﬁg» 3: 5 5% Wii a? mi '9’: 5 :5 m g a g . 1,. ' f ‘ " ' _ {’3 3. Ii Eli is a matrix, then there exist. matrices Q and R such that M m QR, Q has columns which
are orthonormaiigi each other and R is an upper triangular matrix with positive entries on the diagonal. g g:
‘ i .‘M‘t ; ‘ i i » ’ , I i f = “a. j
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AAT m In? Explain. (Hint: consider two cases: 71 = m and n :34 m). (b) {4 points) Is there an orthogonal transformation T from to R3 such that TE) = (a M?) = ($3) if so ﬁnd one: if no: explain Why not. 5:} 3? ifé 21%??? ﬁfiiém 5% i}? 5% mwéié éaé’ é‘ﬁ QM? 2% : if “é'im 5‘ E
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