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# Solutions for Chapter 7.1: Floating-Point Numbers

## Full solutions for Linear Algebra with Applications | 8th Edition

ISBN: 9780136009290

Solutions for Chapter 7.1: Floating-Point Numbers

Solutions for Chapter 7.1
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##### ISBN: 9780136009290

Since 8 problems in chapter 7.1: Floating-Point Numbers have been answered, more than 13038 students have viewed full step-by-step solutions from this chapter. Chapter 7.1: Floating-Point Numbers includes 8 full step-by-step solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This 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: 8.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• 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.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• Hankel matrix H.

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

• Hypercube matrix pl.

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

• 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.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• 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.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• 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.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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).

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.