- 7.1.1: In each of 1 through 4, transform the given equation into a system ...
- 7.1.2: In each of 1 through 4, transform the given equation into a system ...
- 7.1.3: In each of 1 through 4, transform the given equation into a system ...
- 7.1.4: In each of 1 through 4, transform the given equation into a system ...
- 7.1.5: In each of 5 and 6, transform the given initial value problem into ...
- 7.1.6: In each of 5 and 6, transform the given initial value problem into ...
- 7.1.7: Systems of first order equations can sometimes be transformed into ...
- 7.1.8: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
- 7.1.9: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
- 7.1.10: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
- 7.1.11: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
- 7.1.12: In each of 8 through 12, proceed as in 7.(a) Transform the given sy...
- 7.1.13: Transform Eqs. (2) for the parallel circuit into a single second or...
- 7.1.14: Show that if a11, a12, a21, and a22 are constants with a12 and a21 ...
- 7.1.15: Consider the linear homogeneous systemx = p11(t)x + p12(t)y,y = p21...
- 7.1.16: Let x = x1(t), y = y1(t) and x = x2(t), y = y2(t) be any two soluti...
- 7.1.17: Equations (1) can be derived by drawing a free-body diagram showing...
- 7.1.18: Transform the system (1) into a system of first order equations by ...
- 7.1.19: Electric Circuits. The theory of electric circuits, such as that sh...
- 7.1.20: Electric Circuits. The theory of electric circuits, such as that sh...
- 7.1.21: Electric Circuits. The theory of electric circuits, such as that sh...
- 7.1.22: Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1...
- 7.1.23: Consider two interconnected tanks similar to those in Figure 7.1.6....
Solutions for Chapter 7.1: Introduction
Full solutions for Elementary Differential Equations | 10th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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 inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.