 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 and are solutions of the accom...
 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 form a basis for the solution se...
 4.8.6E: Show that the signals in Exercises form a basis for the solution se...
 4.8.7E: In Exercises 7–12, assume the signals listed are solutions of the g...
 4.8.8E: In Exercises, assume the signals listed are solutions of the given ...
 4.8.9E: In Exercises, assume the signals listed are solutions of the given ...
 4.8.10E: In Exercises, assume the signals listed are solutions of the given ...
 4.8.11E: In Exercises, assume the signals listed are solutions of the given ...
 4.8.12E: In Exercises, assume the signals listed are solutions of the given ...
 4.8.13E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.14E: In Exercises find a basis for the solution space of the difference ...
 4.8.15E: In Exercises 13–16, find a basis for the solution space of the diff...
 4.8.16E: In Exercises find a basis for the solution space of the difference ...
 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 form for ...
 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 show that the given signal is a solution of the differ...
 4.8.27E: In Exercises show that the given signal is a solution of the differ...
 4.8.28E: In Exercises show that the given signal is a solution of the differ...
 4.8.29E: Write the difference equations in Exercises and as firstorder syst...
 4.8.30E: Write the difference equations in Exercises and as firstorder syst...
 4.8.31E: Is the following difference equation of order 3? Explain.
 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 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 4.8
Get Full SolutionsThis 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: 9780321982384. Since 37 problems in chapter 4.8 have been answered, more than 40379 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 4.8 includes 37 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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