 6.11.1: Label the following statements as true or false. Assume that the un...
 6.11.2: Prove that rotations, reflections, anel composites of rotations and...
 6.11.3: Let 479 A = ( I 2 v/3 \/3\ 2 1 "2 I and B 0  1 (a) Prove that VA i...
 6.11.4: For any real number , let A = cos* U sm< sin* cos* (a) Prove that L...
 6.11.5: For any real number
 6.11.6: Prove that the composite of any two rotations on R3 is a rotation o...
 6.11.7: Given real numbers (p and ip, define matrices 1 0 0 A =  0 cos
 6.11.8: Prove Theorem 6.45 using the hints preceding the statement of the t...
 6.11.9: Prove that no orthogonal operator can be both a rotation and a refl...
 6.11.10: Prove that if V is a two or threedimensional real inner product s...
 6.11.11: Give an example of an orthogonal operator that is neither a rcfiect...
 6.11.12: Lt;t V be a finitedimensional real inner product spaee. Define T:...
 6.11.13: Complete the proof e>f the lemma to Theorem 6.46 by shewing that W ...
 6.11.14: Let T be an orthogonal [unitary] operator on a finitedimensional r...
 6.11.15: Let T be1 a linear e)perate>r on a finiteeliniensional vector spac...
 6.11.16: Complete the proof of the1 corollary to The'orcm 6.47.
 6.11.17: Let T be' a linear e)perate>r on an ndimensional real inner produc...
 6.11.18: Let V be a real inner product spae*? e)f elimension 2. For any x, y...
Solutions for Chapter 6.11: The Geometry of Orthogonal Operators
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 6.11: The Geometry of Orthogonal Operators
Get Full SolutionsChapter 6.11: The Geometry of Orthogonal Operators includes 18 full stepbystep solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Since 18 problems in chapter 6.11: The Geometry of Orthogonal Operators have been answered, more than 10943 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).