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**Unformatted text preview: **Name: (4.0 {we/- k2,? Math 21b Midterm 2 Tuesday, November 19th, 2002 Please circle your section. Tom Judson Andy Engelward Andy Engelward
Katherine Visnjic (CA) Jakub Topp (CA) Erin Aylward (CA)
MWF 9-10 MWF 10-11 MWF11-12 —-— You have two hours to take this midterm Pace yourself by keeping track of how many problems you
have left to go and how much time remains You don't have to answer the problems in any particular order. So move on to another problem if you ﬁnd you 're stuck and that you are spending too much
time on one problem To receive full credit on a problem, you will need to justify your answers carefully - unsubstantiated
answers, even if correct, will receive little or no credit (except if the directions for that question
speciﬁcally say no justiﬁcation is necessary, such as the True/False). Please be sure to write neatly — illegible answers will also receive little or no credit. If more space is needed, use the back of the previous page to continue your work. Be sure to make a
note of that so that the grader knows where to ﬁnd your answers. You are allowed a half page of notes on it during the test, but you are not allowed to use any other
references or calculators during this test. Good luck! Focus and do well! Question 1. (18 points total) True or False (3 points each) No justiﬁcation is necessary, simply circle T or F for each statement. a F (a) If A is an invertible matrix then the kernels of A and A"1 must be equal.
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.3 «Matt; m2, énkm) s a news mnkCA/gﬂca/s ”,5. Mic gr T ® (c) If a subspace Vof SR3 contains two linearly independent vectors then Vmust contain at
least one of the standard basis vectors as well. <
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zero matrix. , , . . _
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o 1 O O F (e) If A is a (square) orthogonal matrix, and the product AB is also orthogonal, then the
matrix B must be orthogonal as well. “I: 3 a: uanfm'l‘ on 1kg, 41d“ MT M PMJE‘Z op ‘I'wa / arm aA<l M‘t‘l‘u-LQS .5 «(So OF 0M1. No7“. 7347‘ 2 IS Orbs! M
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(wa/urAM=gg=MT - Question 2 (14 points total) Suppose that A = . Find a basis for each of the following subspaces. l 2 3
o 3 3
l l 2
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(c) (2 points) What is the dimension of the kernel (AT)? We. knew 313t- W(AT\=C—va§e—QOC§>LI an) 5.6%.
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s. (lawman a? kiuan (4*) Is A. Question 3. (12 points total) (a) (6 points) Let {5], 172, ..., 17," be a set of vectors that span a subspace V. Suppose W is another vector _. in the subspace V. Show that the set of vectors W53, ,172,...,vm is linearly dependent. __L Sykcc- 5". Vi 37"" \/j‘ "linen-x {Chm N {S ix V/
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nonzero consTmT <.. (b) (6 points) Suppose that 561, 562,. ..,5Ek is a set of k linearly independent vectors in SR" . Suppose A is
an nxn invertible matrix, and thatji = Aic'i for i = 1 to k (i.e. 531 = Ailjz = A552,...jk = Aik ).
Show that set of vectors 5?], 552,” .,?k is also linearly independent. \ SMPPOSC. ’tkc, $2.1, 3" ﬂk’ "'J jK VJuSAI-z— Ingahij
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WK Question 4. (16 points total) . . , 1 1 1 1
(a) (10 points) Flnd an orthonormal ba51s for the kernel of the matrlx [I 2 3 4 [i i i o _( ‘1
‘ C-‘ .o. l =
(In? r'r-b 0‘ A 3 0‘ D. 3 Sb x, X314.“
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basis ~9- W.{:;]l{:;1 Lxerc. we. 3°! G—tNm-SJWQJT' +oiwt. i o \ \ 0 l 5. L
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3— 'X J— “ '5 Fe ( J 5'; —*t o 3 (b) (6 points) Suppose thatA = BC where B is a 4 by 3 matrix and C is a 3 by 4 matrix. Is it possible
for A to be an invertible matrix? If so, give an example (write down matrices A, B and C). If not, explain why not.
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s. A: g; is Mr ieruo‘ﬁblt. Question 5. (16 points total) (a) (6 points) Write down a basis for the linear space of all skew-symmetric3 x 3 matrices (recall that
A is skew-symmetric if AT = -A), and thus determine the dimension of this space. 5179i” £3 wrii'inj sou-m A jeagr-xl 3x3 Sictd-Sjmm‘l‘N—L m‘i‘rtk. “ look iii: 0 Q E “Q; JL-‘DONJ Lu“ in £70.. BN/.
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he. Tim-<4. Law; “(224 SJmM‘i‘nL MA bag 0 (a 3 l E), 02 0-04 Q a“) so The, JORLASL-cq ,C ’6“: inkcm— Sr“. .5 3 (b) (6 points) Find the dimension of the linear space of all symmetricn x n matrices (recall that A is
symmetric if AT = A) Mow which 0(on ’i‘lae. tjPI.CA( WLV‘ 5C 'hAQ. 5 (2.. g Stain/:44. Hi; (\Kh ms‘i'm'us ‘. o‘u‘j‘u ' " qua Moi-L cwnfklxl in, w ’1“; Mam
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JUL‘MW‘E- ‘ SWSL‘ “V“ YDMMV“ ”‘ﬁ‘x \a 3. 11-1: equals 11». Jimenhén. (c) (4 points) If A is an n x n symmetric matrix, then is A2 necessarily symmetnc as well? Explain
why or why not. SMFC. - - - we, (.Arx (Acck Riﬂe M1. US$13 CtMpoM/‘TS : Question 6. (10 points total) Use the method of least squares to ﬁnd the linear function y = mx + b that best ﬁts the following data: y-1216 £74133 TR. 477: “gr J93 ‘lb 71:. equdtln j=MX+L 3mg u: £9!" qum‘lxoms‘. _(=M(—6)+lb ~é l “I
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,5 3... B~X+§x Question 7. (14 points total) (a) (10 points) Let T: V——> Vbe a linear transformation, where Vis a linear space. Suppose that
kemel(Tz) = kemel(T). Show that it must also be the case that kemel(T3) = kemel(Tz). As before, a good strategy is to ﬁrst show that kemel(Tz) g kemel(T3), and then show that kemel(T3) g kemel(T2). (Note, Vis a general linear space, so don't assume that Vis equal to ER“ in
your answer). pc’f‘ 34‘0“) ku- (T‘A‘) é ku— (Te) . I? X é It?» (TA) ‘l'l-tvx
TVxFO, La— The“ TKCX) = 7(Txlx» = TCO'FO J
So X 6 ker’ (T3) <5 we,“ . So kw” (T\) $- LCU" (1—3) (\k‘d- S‘L‘W‘ ker‘éTBJ éku‘éﬂ‘), This .1: LthcP/
Slaw-1- u-‘fk an element“ x .5— kc" LT"), So Ts @‘lzo.
L—Cf‘ TBKKB szZ'l/CKX- Here‘s wiser; TM. «ss«mp‘l';o«
0L0“? ker- L1") = leer (1“) am in sham TKO-04)) :0).
M m) e lemme), w wwkkmum, a .HC
chnxb=o fken So Joe: TCFCX» =0/ ya T\(X)‘—=O.
M f xeww) So mm a“ m) =6,
50 xe kernel (Th) / So kemldﬁj g. kerne( 0.x). 71“ WW] 04} :- Puma! 0-3) (b) (4 points) Give an example of a nonzero linear transformation T from ‘33 to SR3 such that
kemel(T2) is equal to kemel(T). E453 “ASWU—u- TCX‘) = E 32 / 6.5 lith-scl (:3) 2: £53
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- Spring '03
- JUDSON
- Differential Equations, Linear Algebra, Algebra, Equations