 8.1.1E: In Exercises 1–4, write y as an affine combination of the other poi...
 8.1.2E: In Exercises 1–4, write y as an affine combination of the other poi...
 8.1.3E: In Exercises 1–4, write y as an affine combination of the other poi...
 8.1.4E: In Exercises 1–4, write y as an affine combination of the other poi...
 8.1.5E: In Exercises 5 and 6, let Note that S is an orthogonal basis for R3...
 8.1.6E: In Exercises 5 and 6, let Note that S is an orthogonal basis for R3...
 8.1.7E:
 8.1.8E: Repeat Exercise 7 when Reference Exercise 7:
 8.1.9E: Suppose that the solutions of an equation Ax = b are all of
 8.1.10E: Suppose that the solutions of an equation Ax = b are all of
 8.1.11E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 8.1.12E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 8.1.13E:
 8.1.14E: Show that if is a basis for R3, then aff is the plane through
 8.1.15E: Let A be an m × n matrix and, given b in Rm, show that the set S of...
 8.1.16E:
 8.1.17E: Choose a set S of three points such that aff S is the plane in R3 w...
 8.1.18E: Choose a set S of four distinct points in R3 such that aff S is the...
 8.1.19E: Let S be an affine subset of Rn, suppose is a linear transformation...
 8.1.20E: Let be a linear transformation, let T be an affine subset of Rm, an...
 8.1.21E: In Exercises 21–26, prove the given statement about subsets A and B...
 8.1.22E: In Exercises 21–26, prove the given statement about subsets A and B...
 8.1.23E: In Exercises 21–26, prove the given statement about subsets A and B...
 8.1.24E: In Exercises 21–26, prove the given statement about subsets A and B...
 8.1.25E: In Exercises 21–26, prove the given statement about subsets A and B...
 8.1.26E: In Exercises 21–26, prove the given statement about subsets A and B...
Solutions for Chapter 8.1: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 8.1
Get Full SolutionsChapter 8.1 includes 26 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 8.1 have been answered, more than 43309 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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