 7.1: Solve the following lower triangular system of equations. 2x1 = 4 3...
 7.2: Let A = LU. The sequence of transformations R2  RI> R3 +RI> R3  4...
 7.3: Solve the following system of equations using LU decomposition. 6x1...
 7.4: Let A = LU. If you are given L and U how many arithmetic operations...
 7.5: Fmd the condit10n number of the matnx 125 201 Let this matrix be th...
 7.6: Find the condition numbers of the matrices () [i ] 1 1 1 0 0 0 0...
 7.7: Prove that c(A) = c(A1 ).
 7.8: Does c(A) define a linear mapping of Mnn R?
 7.9: Solve the following systems of equations using pivoting and scaling...
 7.10: Solve the following system of equations using the GaussSeidelmethod...
 7.11: Use the power method to determine the dominant eigenvalue and corre...
 7.12: Use the power method with inflation to find all eigenvalues and eig...
 7.13: Find a singular value decomposition of the following matrix A = [ ...
 7.14: LetA be an m X n matrix and P be an m X m orthogonal matrix. Show t...
 7.15: Find the unique minimum length least squares solution of the follow...
Solutions for Chapter 7: Numerical Methods
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 7: Numerical Methods
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 7: Numerical Methods includes 15 full stepbystep solutions. Since 15 problems in chapter 7: Numerical Methods have been answered, more than 8482 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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