 7.9.1: In each of 1 through 12 find the general solution of the given syst...
 7.9.2: In each of 1 through 12 find the general solution of the given syst...
 7.9.3: In each of 1 through 12 find the general solution of the given syst...
 7.9.4: In each of 1 through 12 find the general solution of the given syst...
 7.9.5: In each of 1 through 12 find the general solution of the given syst...
 7.9.6: In each of 1 through 12 find the general solution of the given syst...
 7.9.7: In each of 1 through 12 find the general solution of the given syst...
 7.9.8: In each of 1 through 12 find the general solution of the given syst...
 7.9.9: In each of 1 through 12 find the general solution of the given syst...
 7.9.10: In each of 1 through 12 find the general solution of the given syst...
 7.9.11: In each of 1 through 12 find the general solution of the given syst...
 7.9.12: In each of 1 through 12 find the general solution of the given syst...
 7.9.13: The electric circuit shown in Figure 7.9.1 is described by the syst...
 7.9.14: In each of 14 and 15, verify that the given vector is the general s...
 7.9.15: In each of 14 and 15, verify that the given vector is the general s...
 7.9.16: Let x = (t) be the general solution of x = P(t)x + g(t), and let x ...
 7.9.17: Consider the initial value problemx = Ax + g(t), x(0) = x0.(a) By r...
 7.9.18: Use the Laplace transform to solve the systemx =2 11 2x +2et3t= Ax ...
Solutions for Chapter 7.9: Nonhomogeneous Linear Systems
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.9: Nonhomogeneous Linear Systems
Get Full SolutionsChapter 7.9: Nonhomogeneous Linear Systems includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Since 18 problems in chapter 7.9: Nonhomogeneous Linear Systems have been answered, more than 12159 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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

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

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