midterm_solutions

midterm_solutions - Problem 1[12 Points(a[2 Points Let A be...

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Unformatted text preview: Problem 1 [12 Points] (a) [2 Points] Let A be a nonsingular n x 71 matrix. Describe the factorization (and the matrices involved in it) that is obtained when Gaussian elimination with partial pivoting is applied to A. PA=LH P 7— Pflrwu‘l’mh'om wxotl’r-f’x l. 1'. Mnle IOMr—‘l—rfomamjm- mm‘l’l’lux M 5 V‘UHSl‘M‘XV‘eQr urpef .-+F:qurzmt M0t+frx (b) [6 Points] Apply Gaussian elimination with partial pivoting to the matrix 0 ol «3 G ‘3 A 7 010]A = 1 0 2 Stamp rows 1 0 O o 1 I (.0 and (3) -3 6 *3 > o 2. l km.) (1) «7 roum- Ag) wa) D l I =Hw (2) + g row (I) ((2.:‘l/3) ’3 (a ~3 “‘7 0 7. l .. \rowl‘s) 4) rowC’s)‘J- WW“) — M o '/ (x3144) O 2— \ 0 O 0 at L: -‘/3i 0 P: 01 o o VL 1 ) l 0 0 (C) [4 Points] Use your result (:)}.>tair1ed in (b) to determine the solution of 14.1: b where “‘3 ‘3 ‘2) C 2 '4’ " L—Vé) 1; I 3 z -— L13 ’4 C2 ' a, 3 3 G3 “6 r3)/’g 2. l x : 3 =9 X : (3—H/L :- O I/Z. '/ \ "—N Problem 2 [8 Points] Let a E R be a parameter. For each a, consider the set U 5a Z {111: 6 R3 u + 21) +311} 2 a }. ‘lU (a) [‘2 Points] Determine all values of a E R for which .311 is a subspace of R3. 3 SK SM‘OSPQClfl 3) 7:) o(:. 092‘063u0 3) X: 0 L‘s ‘l’chfl (914/29, FOJJHM WW6 q SO L’s CuaCeeJ a suéfpace: xego) Ce‘2 ‘9 O =CCM+2V+3wJ a (14+ Z(CV)+3(CW\ =9 axe 0 Similar; 283 e .50 a) X+j 6 50 (b) [4 Points] For all values of (1 obtained in (3) determine a basis for the subspace SQ. 7‘ 3 V‘: -\ 6 SD3V1 : .0 Q SD 0 ‘I > V|> V; L‘s O; koala-{S £W S 30:} (c) [2 Points] For all values of (1 obtained in (a) determine a basis for the orthcgonal complement Si of Sn. Sn}; (1 <2. ’ Problem 3 [6 Points] Let 1 1 —1 2 ‘ v1 : 1 , ‘U-g :: 2 , v3 : v; +- ’U2, U1 : 111 — 217-; +173 —1 1 and consider the subspace S : span{ U1, 'U2, v3, 114 } of R4. (3.) [4 Points] Determine a basis for the subspace S. \/3)=v\+vz V%:V‘-2_VLi-V‘3-: Zv'avé -_—.) SPQMlLV\)‘VZ} : SPQMQV‘)V1/ v3)v\t} 2 S VHVL 0\y~{ L‘Marza (“MWWK+1 1—. Civl “' CZVL : —-C\1~zcZ ,_ 0 5 Qi+cz”o MS) V.) VL (‘3 6L 200(813 ‘90“! S (b) [2 Points] Determine the dimension of the orth(‘)gonal cmnpiement S’L of S'. . .L _ (Km/mg Zoe/{wiR‘f‘OQ/{mg‘lLi-Z-IZ Problem 4 Consider the matrix C(A) : Sloan/.110”) Q3} . 0") £13 aqu owe Knew/(3 Lnabrwohui~ (b) [4 Points] Determine the dimensions of the four fundamental subspaces C/Q/‘i m C/ ) ‘= 2— C; \" ob‘w. NCA) : 3*2. 2| aw CcA') A..— - fault xi+X3 1:0 __ A X :0 C: -x. 4.x} :- Xl ’ T [x‘ _ K x- A O Fh/UAJ‘: N(A): X:Yig]\ X1219} : o \ OWA 22 [S Ct foams {W [14 Points] (J a. «@492: (EN CH) of A. (d) [2 Points] Find a basis for C(AT). ., \ _ T l . i .. l l 2) [ll]) [71;] \‘S a. 901391 £01’ (e) [2 Points] Determine if A has a right-inverse, and if it does, determine all its right- inverses. Aé le’doU/ml rank—A = Z "'9 A has 0L Wald —an’r~£€ Cu C47. ‘ 0 C” i” C3. C|Z+ C37. C3! C32 [0 J: A C : Cut+Csi C|1+C32 _C‘ +C' __0 C3I=lé H 3| (2.9 C -. Ctz+C32=0 "’ /2. V; "'VZ "C12 *‘ (32 =l 6’32 :‘Q C c: _—l D) C 2 C11 '22. I CZ!) C22 QQ 2 - /a V; /L ) (f) [2 Points] Determine if A has a left-inverse and if it does, determine all its left-inverses. _w3 AelR Mi fahuA=z<3 =9 A lnas mo filtd‘mwrse Problem 5 [10 Points] True or False? (a) [2 Points] Gaussian elimination with partial pivoting applied to a nonsingular matrix can break down due to a zero pivot. El rPrue MFalse (b) [2 Points] If a subspace S of a vector space V satisfies 5' :2 Si, then S 2 V 2 MTrue [:1 False (c) [2 Points] A least-squares problem always has at least one solution. [ll/True El False (cl) [2 Points] Any matrix A 6 IR’”” matrix has a right—inverse. El True EFFalse (e) [2 Points] The set of all vectors u ‘U E R3 with w:1 ;: l U) is a subspace of R3. El True M False ...
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midterm_solutions - Problem 1[12 Points(a[2 Points Let A be...

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