 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 4350 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.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).