 7.9.1: In each of 1 through 8 find the general solution of the given syste...
 7.9.2: In each of 1 through 8 find the general solution of the given syste...
 7.9.3: In each of 1 through 8 find the general solution of the given syste...
 7.9.4: In each of 1 through 8 find the general solution of the given syste...
 7.9.5: In each of 1 through 8 find the general solution of the given syste...
 7.9.6: In each of 1 through 8 find the general solution of the given syste...
 7.9.7: In each of 1 through 8 find the general solution of the given syste...
 7.9.8: In each of 1 through 8 find the general solution of the given syste...
 7.9.9: The electric circuit shown in Figure 7.9.1 is described by the syst...
 7.9.10: In each of 10 and 11, verify that the given vector is the general s...
 7.9.11: In each of 10 and 11, verify that the given vector is the general s...
 7.9.12: Let x = _(t) be the general solution of x_ = P(t)x + g(t), and let ...
 7.9.13: In 13 and 14, you are given a nonhomogeneous secondorder linear di...
 7.9.14: In 13 and 14, you are given a nonhomogeneous secondorder linear di...
 7.9.15: Carry out steps a through e for the general nonhomogeneous secondo...
 7.9.16: Consider the initial value problem x _ = Ax + g(t), x(0) = x0. a. B...
 7.9.17: Use the Laplace transform to solve the system x _ = _2 1 1 2 _ x + ...
Solutions for Chapter 7.9: Nonhomogeneous Linear Systems
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 7.9: Nonhomogeneous Linear Systems
Get Full SolutionsChapter 7.9: Nonhomogeneous Linear Systems includes 17 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Since 17 problems in chapter 7.9: Nonhomogeneous Linear Systems have been answered, more than 12334 students have viewed full stepbystep solutions from this chapter.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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).