 6.2.1: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.2: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.3: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.4: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.5: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.6: In Exercises 16, compute the first three iterations of the PowerMet...
 6.2.7: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.8: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.9: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.10: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.11: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.12: In Exercises 712, compute the first two iterations of the PowerMeth...
 6.2.13: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.14: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.15: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.16: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.17: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.18: In Exercises 1318, the eigenvalues of a 3 3 matrix A are given.Dete...
 6.2.19: In Exercises 1922, the given is the dominant eigenvalue for A.To wh...
 6.2.20: In Exercises 1922, the given is the dominant eigenvalue for A.To wh...
 6.2.21: In Exercises 1922, the given is the dominant eigenvalue for A.To wh...
 6.2.22: In Exercises 1922, the given is the dominant eigenvalue for A.To wh...
 6.2.23: In Exercises 2326, to which matrix B would you apply the InversePow...
 6.2.24: In Exercises 2326, to which matrix B would you apply the InversePow...
 6.2.25: In Exercises 2326, to which matrix B would you apply the InversePow...
 6.2.26: In Exercises 2326, to which matrix B would you apply the InversePow...
 6.2.27: Below is the output resulting from applying the Inverse PowerMethod...
 6.2.28: Below is the output resulting from applying the Shifted InversePowe...
 6.2.29: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.30: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.31: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.32: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.33: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.34: FIND AN EXAMPLE For Exercises 2934, find an example thatmeets the g...
 6.2.35: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.36: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.37: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.38: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.39: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.40: TRUE OR FALSE For Exercises 3540, determine if the statementis true...
 6.2.41: For the matrix A and vector x0, compute the first four iterationsof...
 6.2.42: For the matrix A and vector x0, compute the first four iterationsof...
 6.2.43: For the matrix A, = 2 is the largest eigenvalue. Use the givenvalue...
 6.2.44: For the matrix A, = 3 is the largest eigenvalue. Use the givenvalue...
 6.2.45: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.46: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.47: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.48: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.49: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.50: C In Exercises 4550, compute the first six iterations of the PowerM...
 6.2.51: C In Exercises 5156, compute as many iterations of the PowerMethod ...
 6.2.52: C In Exercises 5156, compute as many iterations of the PowerMethod ...
 6.2.53: C In Exercises 5156, compute as many iterations of the PowerMethod ...
 6.2.54: C In Exercises 5156, compute as many iterations of the PowerMethod ...
 6.2.55: C In Exercises 5156, compute as many iterations of the PowerMethod ...
 6.2.56: C In Exercises 5156, compute as many iterations of the PowerMethod ...
Solutions for Chapter 6.2: Approximation Methods
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 6.2: Approximation Methods
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. Chapter 6.2: Approximation Methods includes 56 full stepbystep solutions. Since 56 problems in chapter 6.2: Approximation Methods have been answered, more than 14514 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Outer product uv T
= column times row = rank one matrix.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.