- 8.4.1: Consider the linear system of differential equations (3) Find the g...
- 8.4.2: Consider the linear system of differential equations o - 2 o (3) Fi...
- 8.4.3: Find the general solution to the linear system of differenlial equa...
- 8.4.4: Prove that the set of all solutions to the homogeneous linear syste...
- 8.4.5: Find the general solution to the linear system of differential equa...
- 8.4.6: Find the general solution to the linear system of differenlial equa...
- 8.4.7: Find the general solution to the linear system of differential equa...
- 8.4.8: Find the generul solution 10 the linear system of differen tial equ...
- 8.4.9: Consider two competing species that live in the same forest. and le...
- 8.4.10: Suppose that we have a system consisting of two interconnected tank...
Solutions for Chapter 8.4: Differential Equations
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Remove row i and column j; multiply the determinant by (-I)i + j •
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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).
Every v in V is orthogonal to every w in W.
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.
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.
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 = Q-1 = Q.
Solvable system Ax = b.
The right side b is in the column space of A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.