 2.5.1E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.2E: In Exercises, solve the equation Ax = b by using the LU factorizati...
 2.5.3E: In Exercises, solve the equation Ax = b by using the LU factorizati...
 2.5.4E: In Exercises, solve the equation Ax = b by using the LU factorizati...
 2.5.5E: In Exercises, solve the equation Ax = b by using the LU factorizati...
 2.5.6E: In Exercises, solve the equation Ax = b by using the LU factorizati...
 2.5.7E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.8E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.9E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.10E: Find an Lu factorization of the matrices in Exercises (with ith L u...
 2.5.11E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.12E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.13E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.14E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.15E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.16E: Find an Lu factorization of the matrices in Exercises (with L unit ...
 2.5.17E: When A is invertible, MATLAB finds A –1 by factoring A = LU (where ...
 2.5.18E: Find A–1 as in Exercise 17, using A from Exercise 3.Exercise 17:Whe...
 2.5.19E: Let A be a lower triangular n × n matrix with nonzero entries on th...
 2.5.20E: Let A = LU be an LU factorization. Explain why A can be row reduced...
 2.5.21E: Suppose A = BC, where B is invertible. Show that any sequence of ro...
 2.5.22E: Exercises 22–26 provide a glimpse of some widely used matrix factor...
 2.5.23E: Exercises 22–26 provide a glimpse of some widely used matrix factor...
 2.5.24E: Exercises 22–26 provide a glimpse of some widely used matrix factor...
 2.5.25E: Exercises 22–26 provide a glimpse of some widely used matrix factor...
 2.5.26E: Exercises provide a glimpse of some widely used matrix factorizatio...
 2.5.27E: Design two different ladder networks that each output 9 volts and 4...
 2.5.28E: Show that if three shunt circuits (with resistances (R1, R2, R3) ar...
 2.5.29E: a. Compute the transfer matrix of the network in the figure.b. Let ...
 2.5.30E: Find a different factorization of the transfer matrix A in Exercise...
 2.5.31E: [M] Consider the heat plate in the following figure (refer to Exerc...
 2.5.32E: [M] The band matrix A shown below can be used to estimate the unste...
Solutions for Chapter 2.5: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 2.5
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Chapter 2.5 includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Since 32 problems in chapter 2.5 have been answered, more than 40592 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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.

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].

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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!

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

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

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).