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Math 2270 Assignment 9 - Dylan Zwick Fall 2012 Section 4.3 1, 3, 9, 10, 12 Section 4.4 2, 6, 11, 18, 21 Section 5.1 1, 3, 8, 13, 24 - - - 1 4.3 Least Squares Approximation - 4.3.1 ith b = 0, 8. 8. 20 at t = 0, 1. 3, 4, set up and solve the normal equa ATb. For the best straight line find its four heights p tions ATA$ and four errors c. What is the minimum value E = c + c + e + c? ATA/’ ( (‘o’Jq 3 H (L ATJ, i ui (o3)/ {J (X( i I1L - l6J (g +ZLlI - () 17 (III H (Ii) 2 E (/ / I A I ) 3 ( 5) 41) 17 Al i 4.3.3 Check that e = b—p = (—1, 3, —5, 3) is perpendicular to both columns of A. What is the shortest distance from b to the column space of A? 11) /i I (ô O+3-i5±f3= 1Ief( ILS)l+ 3 2 to the same four points, 4.3.9 For the closest parabola b = C + D1 + Et write down the unsolvable equations Ax = b in three unknowns ATb (solution (C, D, E). Set up the three normal equations ATA* not required). C 7tij 1 o 13 1 9 ( I o o\ / c1 39 i ( i ( 26 I I /\ H If OI3 J/ 1 OH(1CJ/ (?cJ AA 9 A 7 r/ ( I () 4.3.10 For the closest cubic b = C + Dt + Et 2 + Ft 3 to the same four points, write down the four equations Ax = b. Solve them by elimination. What are p and e? /o0 C /1111 I 3 L7 i I6 I I 4 19 (coo 01 0 C) C) /10 / 1 q7 3 f 3 ) F 0 —S 3 i—i - 6L{ IC 0 L Lc) - L 6 3 pj Lj7 (o IC 3 3_ j9 (z b F I 1;; 5 / 3 ,. , b,,,) onto the 1 (b 4.3.12 (Recommended) This problem projects b b in 1 Un line through a = (1,. , 1). We solve m equations ar known (by least squares). . (a) Solve b’s. aTa± (b) Find e = b aTb — to show that § is the mean (the average) of the a and the variance eU 2 and the standard deviation (3, 3, 3) is closest to b = (1, 2, 6). Check (c) The horizontal line 1 that p (3, 3, 3) is perpendicular to e and find the 3 by 3 projec tion matrix P. t (/ 1)’ I ((I J 47) e ( /:) / : /) 6 c) -1 3 O 31 \. ‘ 4.4 Orthogonal Bases and Gram-Schmidt - 4.4.2 The vectors (2, 2, —1) and (—1, 2, 2) are orthogonal. Divide them by their lengths to find orthonormal vectors q 1 and q . Put those into 9 the columns of Q and multiply QTQ and QQT. z(a) 317/ ff1 C) or4 k “1 show that their product 4.4.6 If Q 1 and Q2 are orthogonal matrices, QTQ = I.) is also an orthogonal matrix. (Use A- 41 Q1Q2 / -7 (QiLT(Q, L] 1 ,TQ v 8 4.4.11 (a) Gram-Schmidt: Find orthonormal vectors q 1 and q 9 in the plane spannedbya = (1, 3,4, 5, 7) andb = (—6,6,8,0,8). (b) Which vector in this plane is closest to (1, 0, 0, 0, 0)? ) -7\/ 1 IOC 3 9 J T 4 ‘9 ) L joo(\; 57 j (H 1 $ 3LH51 —1 0 I — 5 - fcc) - 4.4.18 (Recommended) Find orthogonal vectors A, B, C by Gram-Schmidt from a, b, C: a = (1, —1,0,0) b=(0,1,—1,O) N - I2i- /lCAVt. 7_i (!) A U C) j3 I — c=(0,O,i,—1). Find an orthonormal basis for the column space of A: 1 —2 10 11 13 A and b=(3). Then compute the projection of b onto that column space. 1 A mJ 0 M 7 zJiT 7’ —9 /tJii R f L L L5 LT 4* - It \\ I S 32’fiT \Z 11 ( - 5.1 The Properties of Determinants - 5.1.1 If a 4 x 4 matrix has dct(A) , find dt(2A) and dt(—A) and + ) IAH) e4) 12 5.1.3 True of false, with a reason if true and a couriterexample if false: (a) The determinant of I + A is 1 + dt(A). (b) The determinant of ABC is ABHC. (c) The determinant of 411 is (d) The determinant of AB (0 0 1 — BA is zero. Try an example with A -I 11 I u11I,c I e I ff 4-I) (,4 : - d fTh 4 cr’) 1 F J / / C (C)t /0(1 Gi / ‘4= (f;) 0 O () 13 50) rnq ‘ /‘ CL fo-/) 5.1.8 Prove that every orthogonal matrix —1. (QTQ I) has determinant 1 or A B and the transpose rule (a) Use the product rule ABI Q = L T Q (b) Use only the product rule. If dct(Q) > 1 then dt(Q) dct(Q) blows up. How do you know this can’t happen to Q? Qi (o&H 1&HLQ/ 11 12D 50 Pc n J1 ‘y 00 k jf rio// 14 I_/f educe A to U and find dct(A) product of the pivots: /1 1 i\ A( 122) 1 2 3) /1 2 3 A=( 223 \3 3 3 () t() —) (o tIJ (A) HP / [RJ 7) 3 33/ /o- /1 ) I Z O-3 O (A / 15 (-i) (-) 5.1.24 Elimination reduces A to U. Then A /1 /33 4’\ Arr(687)=12 -1 5 -9) = LU: OO’\/33 4\ 1O)f02_1)LU. 4 1) \O 0 -1) L’A. 1 Find the determinants of L, U, A, U’L’, and U () () AH ILl lL’/ ((-) + (u ‘j / A4- (t’) (u ‘L ‘ d+ o -y4-) dfF) 16