 7.1.1: Find the threedigit decimal floatingpoint representation of each ...
 7.1.2: Find the absolute error and the relative error when each of the rea...
 7.1.3: Represent each of the following as fivedigit base 2 floatingpoint...
 7.1.4: Use fourdigit decimal floatingpoint arithmetic to do each of the ...
 7.1.5: Let x1 = 94,210, x2 = 8631, x3 = 1440, x4 = 133, and x5 = 34. Calcu...
 7.1.6: What would the machine epsilon be for a computer that uses 16digit...
 7.1.7: What would the machine epsilon be for a computer that uses 36digit...
 7.1.8: How many floatingpoint numbers are there in the system if t = 2, L...
Solutions for Chapter 7.1: FloatingPoint Numbers
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7.1: FloatingPoint Numbers
Get Full SolutionsSince 8 problems in chapter 7.1: FloatingPoint Numbers have been answered, more than 13038 students have viewed full stepbystep solutions from this chapter. Chapter 7.1: FloatingPoint Numbers includes 8 full stepbystep 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.

CayleyHamilton 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 Fn1c 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 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.

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. AI is also symmetric.