 7.5.1: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.2: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.3: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.4: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.5: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.6: In each of 1 through 6:(a) Find the general solution of the given s...
 7.5.7: In each of 7 and 8:(a) Find the general solution of the given syste...
 7.5.8: In each of 7 and 8:(a) Find the general solution of the given syste...
 7.5.9: In each of 9 through 14, find the general solution of the given sys...
 7.5.10: In each of 9 through 14, find the general solution of the given sys...
 7.5.11: In each of 9 through 14, find the general solution of the given sys...
 7.5.12: In each of 9 through 14, find the general solution of the given sys...
 7.5.13: In each of 9 through 14, find the general solution of the given sys...
 7.5.14: In each of 9 through 14, find the general solution of the given sys...
 7.5.15: In each of 15 through 18, solve the given initial value problem. De...
 7.5.16: In each of 15 through 18, solve the given initial value problem. De...
 7.5.17: In each of 15 through 18, solve the given initial value problem. De...
 7.5.18: In each of 15 through 18, solve the given initial value problem. De...
 7.5.19: The system tx = Ax is analogous to the second order Euler equation ...
 7.5.20: Referring to 19, solve the given system of equations in each of 20 ...
 7.5.21: Referring to 19, solve the given system of equations in each of 20 ...
 7.5.22: Referring to 19, solve the given system of equations in each of 20 ...
 7.5.23: Referring to 19, solve the given system of equations in each of 20 ...
 7.5.24: In each of 24 through 27, the eigenvalues and eigenvectors of a mat...
 7.5.25: In each of 24 through 27, the eigenvalues and eigenvectors of a mat...
 7.5.26: In each of 24 through 27, the eigenvalues and eigenvectors of a mat...
 7.5.27: In each of 24 through 27, the eigenvalues and eigenvectors of a mat...
 7.5.28: Consider a 2 2 system x = Ax. If we assume that r1 = r2, the genera...
 7.5.29: Consider the equationay + by + cy = 0, (i)where a, b, and c are con...
 7.5.30: The twotank system of in Section 7.1 leads to the initial value pr...
 7.5.31: Consider the systemx =1 1 1x.(a) Solve the system for = 0.5. What a...
 7.5.32: Electric Circuits. 32 and 33 are concerned with the electric circui...
 7.5.33: Electric Circuits. 32 and 33 are concerned with the electric circui...
Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Since 33 problems in chapter 7.5: Homogeneous Linear Systems with Constant Coefficients have been answered, more than 12201 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients includes 33 full stepbystep solutions.

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

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.