A6_soln

# A6_soln - Math 235 Assignment 6 Solutions 1. Find a and b...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 235 Assignment 6 Solutions 1. Find a and b to obtain the best ﬁtting equation of the form y = a + bt for the given data. t 1 2 3 4 5 y 9 6 5 31 9 11 6 12 Solution: We take ﬂ 2: 5 andX = 1 3 .Then 3 1 4 1 1 5 9 ~1 6 a_T-1T,,__515 11111 __1~_55~1524*10.5 [61‘(XX) X “[1555] 1123451§“50 ~15 5 53”~1.9 1 Hence, the best ﬁtting line is y = 10.5 ~ 1.916. 2. Verify that the following system A2? =2 5 is inconsistent, then determine for the vector :3 that minimizes [[1455 ~ £131 ‘1‘ 21'2 Z 5 2961 ~ 3332 2: 6 £191 "‘ 12112 I: “*4 1 2 5 1 2 5 Solution: Row reducing gives 2 ~13 6 N 0 ~7 ~41 Hence, the system is 1 ~12 «4 0 0 ~1 inconsistent. 1 2 .. We have A = 2 ~23 . Hence, the vector :5 that minimizes “As? —— [9H is 1 ~12 f:(ATA)“1A5 “ 6 ~16 "1 1 2 1 g “ ~16 157 2 ~3 ~12 __4 ____1__ 157 16 13 “685 16 6 40 2: [3385/4998] 2 3. Prove that if U and W are subspaces of a vector space V such that U 0 W = {6}, then U 63 W is a subspace of V. Moreover, if {171, . . . ,ﬂk} is a basis for U and {2171,- -- ,w’g} is a basis for W, then {271, . . . £19,131, . . . ,u‘ig} is a basis for U EB W. Solution: We Will prove that {171, . . . ,17k+g} is a linearly independent spanning set for UEEW. Consider C1111 + - - - +Ck’17k + 0194,1131 + ' ' ‘ + Ck+gtvg == 0 Then, we have 61771 + ' ' ' + Ck'Uk : “CMﬂBi '* ' ' ' ‘“ Ck+€117£ The vector on the right is in U and the vector on the left is in W. But, the only vector that is both in U and W is the zero vector. Therefore, each oz- 2 O and hence {171, . . . , 17k, 21517 . . . , 117g} is linearly independent. For any 17 E U EB W we have that 17 =2 {i + ’u')’ by deﬁnition of U G9 W. We can write 11’: 01171 + ' ' ‘ + Ckﬂk and U7: d1131+~~+ dew} and hence 17201771+"'+Ck’17k+d1231+"‘+dg’tvg and so {171,...,17k,161,...,tb’g} also spans UEBW. 4. Let U be a subspace of a ﬁnite dimensional vector space V and suppose that there are subspaces W1 and W2 such that V = U @ W1 and V = U EB W2. (a) Prove that dim W1 = dim W2. Solution: By Theorem 3.5.1 (problem 3), we have that dimV 2: dimU + dile and dimV :2 dimU + dim W2. Thus, dim W1 :2 dimV —— dimU =2 dim W2 (b) Give an example of a vector space V With subspaces U, W1, and W2 such that V = UEBW1 andV—TUEBWQ, but W1 Solution: In R2, let U = Span { }, W1 :2 Span{ } and W2 2 Span{ Then, by Theorem 3.5.1, we have V 2 U EB W1 and V = U 89 W2, but clearly W1 % W2. 5. Let A be an m x 71 real matrix. Prove that the nullspace of AT is the orthogonal complement of the column space of A. 17? Solution: Let A = [171 17”], so AT :2 _ 5:5 Ufa: HfEQﬂAHJMnQdHJLEmmmATH= ; zﬁhmmfeNmmﬂy 173% On the other hand, let :3 E Null(AT), then AT” :2 6 so 13} - a? 2: 0 for all 2'. Now, pick any 56 Col(A). Then 5: 01171 + - ~ - + cuff”. But then szkﬁﬁ~~+%%yfzm hence a? E Col(A)i. MATH 235 Assignment 6 MATLAB Solutions Least Squares (a) X = [0; 10; 20; 30; 40; X = 0 10 20 30 40 50 60 70 80 90 y = [4.77; 16.47; 19.76; y = 4.7700 16.4700 19.7600 22.3500 23.6700 25.0300 27.0100 30.9300 40.9800 53.8600 plOt(XI Y! H”); (b) X = [x X.“2 x.“3] X 7:. 0 O 10 100 20 400 30 900 40 1600 50 2500 60 3600 70 4900 80 6400 90 8100 50; 60; 70; 22.35; 23.67; 0 1000 8000 27000 64000 125000 216000 343000 512000 729000 80; 90] 25.03; 27.01; 30.93; 40.98; 53.86] rref(X'*X) ans = l O O 0 1 0 0 0 1 beta = inV(X'*X)*X'*y beta = 1.6288 —0.0366 0.0003 (d) xnrange = [0: l: 901'; ymfitnline = beta(l,l)*x_range + beta(2,l)*x_range.“2 + beta(3,1)*x_range.“3; hold on plot(x_range, y_fit_line, 'r—') MATH 235 Mm fa: Assignment 5 £933: Sqaams: Safes v3. Costs costs (méltimg) wk 5313 minis WWW i235! sqeareg {it {I E} 2:3 33 413 Sit 80 m 833 238 sales (méiﬁtms) ...
View Full Document

## This note was uploaded on 09/30/2011 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

### Page1 / 5

A6_soln - Math 235 Assignment 6 Solutions 1. Find a and b...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online