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# 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

Solutions for Chapter 8.1
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##### ISBN: 9780321982384

Chapter 8.1 includes 26 full step-by-step 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 step-by-step 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.

Key Math Terms and definitions covered in this textbook
• 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 S-I 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 A-I.

Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. 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 = time-invariant (shift-invariant) 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).

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