 4.8.1E: Verify that the signals in Exercises 1 and 2 are solutions of the a...
 4.8.2E: Verify that the signals in Exercises 1 and 2 are solutions of the a...
 4.8.3E: Show that the signals in Exercises 3–6 form a basis for the solutio...
 4.8.4E: Show that the signals in Exercises 3–6 form a basis for the solutio...
 4.8.5E: Show that the signals in Exercises 3–6 form a basis for the solutio...
 4.8.6E: Show that the signals in Exercises 3–6 form a basis for the solutio...
 4.8.7E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.8E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.9E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.10E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.11E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.12E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.13E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.14E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.15E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.16E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.17E: Exercises 17 and 18 concern a simple model of the national economy ...
 4.8.18E: Exercises 17 and 18 concern a simple model of the national economy ...
 4.8.19E: A lightweight cantilevered beam is supported at N points spaced 10 ...
 4.8.20E: A lightweight cantilevered beam is supported at N points spaced 10 ...
 4.8.21E: When a signal is produced from a sequence of measurements made on a...
 4.8.22E: Let be the sequence produced by sampling the continuous signal as s...
 4.8.23E: Exercises 23 and 24 refer to a difference equation of the formyk + ...
 4.8.24E: Exercises 23 and 24 refer to a difference equation of the form for ...
 4.8.25E: In Exercises 25–28, show that the given signal is a solution of the...
 4.8.26E: In Exercises 25–28, show that the given signal is a solution of the...
 4.8.27E: In Exercises 25–28, show that the given signal is a solution of the...
 4.8.28E: In Exercises 25–28, show that the given signal is a solution of the...
 4.8.29E: Write the difference equations in Exercises 29 and 30 as firstorde...
 4.8.30E: Write the difference equations in Exercises 29 and 30 as firstorde...
 4.8.31E: Is the following difference equation of order 3? Explain.yk + 3 + 5...
 4.8.32E: What is the order of the following difference equation? Explain you...
 4.8.33E: Are the signals and linearly independent? Evaluate the associated C...
 4.8.34E: Let f , g, and h be linearly independent functions defined for all ...
 4.8.35E: Must the signals be linearly independent in S? Discuss.Let a and b ...
 4.8.36E: Must the signals be linearly independent in S? Discuss.Let V be a v...
 4.8.37E: Must the signals be linearly independent in S? Discuss.Let S0 be th...
Solutions for Chapter 4.8: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.8
Get Full SolutionsSince 37 problems in chapter 4.8 have been answered, more than 32241 students have viewed full stepbystep solutions from this chapter. Chapter 4.8 includes 37 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.