Hw9_soln

Hw9_soln - 4 8 5.3 Math 27‘ 19W \$8 I6,2-| 2 35 3,39H0 42...

This preview shows pages 1–2. Sign up to view the full content.

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: 4. 8. 5.3 Math 27‘] 19W? \$8, I6,2'-|, 2%. 35.. 3?,39H0, 42 Not orthogonal, the ﬁrst and third column vectors fail to be perpendicular to each other. A + B will not necessarily be orthogonal, because the columns may not be unit vectors. For example, if A = B = n, then A + B = 2L“, which is not orthogonal. 16. A + B is symmetric, since (A + B)T : AT + BT = A + B.- 24. Not necessarily symmetric. (ATBA)T = AT (ATB)T = ATBTA. 28. Write L(:E) = A55; by Deﬁnition 5.3.1, A is an orthogonal n x 71 matrix, so that ATA : In, 35. by Fact 5.3.7. Now L(17)-L(1Ei) = (Am-(A13) = (At—OTA'LD' =17TATA1IJ' = “JFIH'LU 2 v u? =— 17 - If}, as claimed. Note that we have used Facts 5.3.6 and 5.3.93. Let us ﬁrst think about the inverse L = T—1 of T. 2 . _, _, _, _, _. _' _ 5 Write Lt”) = A113 = [111 112 v3 ] :8. It 13 required that L(é'3) : {1‘3 = g .1. 3 Furthermore, tilegvectors 111, v2, '03 must form an orthonormal basis of R3. By inspection, 3 we ﬁnd 171 = % 2 3 1 ‘5 —2 —1 2 Then 172 2 171x63 = g does thejob. In summary, we have LU?) : § 1 2 2 5.". _§ 2 —2 1 Since the matrix of L is orthogonal, the matrix of T = L” is the transpose of the matrix of L: H2 1 2 T(;E)=% —1 2 —2 5;“ 2 2 1 There are many other answers (since there are many choices for the vector 171 above). 2 —3 3 2 37. No, since the vectors 3 and 2 are orthogonal, whereas 0 and —3 are not 0 0 2 0 (see Fact 5.3.2). 39. By Fact 5.3.10, the matrix of the projection is WIT; the ijth entry of this matrix is ui'uj. 0.5 -—0.1 40 An ortho l " ' ” — 0'5 " 0'7 ' . norina basis of W IS ul — 0 5 ,U2 = _0 7 (see Exercrse 5.2.9). 0.5 0.1 By Fact 5.3.10, the matrix of the projection onto W is QQT, where Q = [1'51 172 26 18 32 24 1s 74 —24 32 T__1_ QQ ‘100 32 -24 74 18 24 32 18 26 42. a. Suppose we are projecting onto a subspace W of 13.“. Since Arif is in W already, the orthogonal projection of Ari? onto W is just Ai" itself: A(A:E) : Af, or A2 " = Airf. Since this equation holds for all if, we have A2 = A. b. A = QQT, for some matrix Q with orthonormal columns 111, . . . , 11m. Note that QTQ = 1..., since the ijth entry of QTQ is n;- - 113-. Then A2 = QQTQQT = Q(QTQ)QT = QImQT = QQT = A- ...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

Hw9_soln - 4 8 5.3 Math 27‘ 19W \$8 I6,2-| 2 35 3,39H0 42...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online