 2.5.1E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.2E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.3E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.4E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.5E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 2.5.6E: In Exercises 1–6, solve the equation Ax = b by using the LU factori...
 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 7–16 (with L ...
 2.5.9E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.10E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.11E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.12E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 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 7–16 (with L ...
 2.5.15E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 2.5.16E: Find an LU factorization of the matrices in Exercises 7–16 (with L ...
 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 22–26 provide a glimpse of some widely used matrix factor...
 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 below. ...
 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 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 2.5
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Chapter 2.5 includes 32 full stepbystep solutions. Since 32 problems in chapter 2.5 have been answered, more than 32467 students have viewed full stepbystep solutions from this chapter. 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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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 ().

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.