 7.1: Set A = round(10 rand(6)) s = ones(6, 1) b = A s The solution of th...
 7.2: If a vector y Rn is used to construct an n n Vandermonde matrix V, ...
 7.3: Construct a matrix C as follows: Set A = round(100 rand(4)) L = tri...
 7.4: The n n Hilbert matrix H is defined by h(i, j ) = 1/(i + j 1) i, j ...
 7.5: Use MATLAB to compute the eigenvalues and eigenvectors of a random ...
 7.6: Set A = round(10 rand(5)); A = A + A_ [X, D] = eig(A) Compute cond(...
 7.7: Set A = magic(4) and t = trace(A). The scalar t should be an eigenv...
 7.8: Set A = diag(10 : 1 : 1) + 10 diag(ones(1, 9), 1) [X, D] = eig(A) C...
 7.9: Construct a matrix A as follows: A = diag(11 : 1 : 1,1); for j = 0 ...
 7.10: (a) In MATLAB, the simplest way to compute a Householder matrix tha...
 7.11: Set x1 = (1 : 5) _; x2 = [1, 3, 4, 5, 9]_; x = [x1; x2] Construct a...
 7.12: To plot y = sin(x), we must define vectors of x and y values and th...
 7.13: Let A = 4 5 2 4 5 2 0 3 6 0 3 6 Enter the matrix A in MATLAB and co...
 7.14: Set A = round(10 rand(10, 5)) and s = svd(A) (a) Use MATLAB to comp...
 7.15: Set A = rand(8, 4) rand(4, 6), [U, D, V ] = svd(A) (a) What is the ...
 7.16: With each A Rnn, we can associate n closed circular disks in the co...
 7.17: We can generate a random symmetric 10 10 matrix by setting A = rand...
 7.18: A nonsymmetric matrix A may have complex eigenvalues. We can determ...
 7.19: (a) Generate 100 random 55 matrices and compute the condition numbe...
Solutions for Chapter 7: Numerical Linear Algebra
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7: Numerical Linear Algebra
Get Full SolutionsSince 19 problems in chapter 7: Numerical Linear Algebra have been answered, more than 7495 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 7: Numerical Linear Algebra includes 19 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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