- 10.18.1: Let a certain animal population be divided into three 1-year age cl...
- 10.18.2: For the optimal sustainable harvesting policy described by Equation...
- 10.18.3: Use Equation 10 to show that if only the first age class of an anim...
- 10.18.4: If only the ith class of an animal population is to be periodically...
- 10.18.5: Suppose that all of the Jth class and a certain fraction of the Ith...
- 10.18.T1: The results of Theorem 10.18.1 suggest the following algorithm for ...
- 10.18.T2: Using the algorithm in Exercise T1 , do the one-age-class calculati...
- 10.18.T3: Referring to the mouse population in Exercise T3 of Section 10.17, ...
Solutions for Chapter 10.18: Harvesting of Animal Populations
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
peA) = det(A - AI) has peA) = zero matrix.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Invert A by row operations on [A I] to reach [I A-I].
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.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = 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.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.