 7.5.1: 14 Sketch the curve by using the parametric equations to plot point...
 7.5.2: 14 Sketch the curve by using the parametric equations to plot point...
 7.5.3: 14 Sketch the curve by using the parametric equations to plot point...
 7.5.4: 14 Sketch the curve by using the parametric equations to plot point...
 7.5.5: 58 (a) Sketch the curve by using the parametric equations to plot p...
 7.5.6: 58 (a) Sketch the curve by using the parametric equations to plot p...
 7.5.7: 58 (a) Sketch the curve by using the parametric equations to plot p...
 7.5.8: 58 (a) Sketch the curve by using the parametric equations to plot p...
 7.5.9: 912 Describe the motion of a particle with position sx, yd as t var...
 7.5.10: 912 Describe the motion of a particle with position sx, yd as t var...
 7.5.11: 912 Describe the motion of a particle with position sx, yd as t var...
 7.5.12: 912 Describe the motion of a particle with position sx, yd as t var...
 7.5.13: Predatorprey equations For each predatorprey system, determine wh...
 7.5.14: Competition and cooperation Each system of differential equations i...
 7.5.15: Cooperation, competition, or predation? The system of differential ...
 7.5.16: A food web Lynx eat snowshoe hares, and snowshoe hares eat woody pl...
 7.5.17: 1718 Rabbits and foxes A phase trajectory is shown for populations ...
 7.5.18: 1718 Rabbits and foxes A phase trajectory is shown for populations ...
 7.5.19: 1920 Graphs of populations of two species are shown. Use them to sk...
 7.5.20: 1920 Graphs of populations of two species are shown. Use them to sk...
 7.5.21: LotkaVolterra equations In Example 1(a) we showed that parametric ...
 7.5.22: LotkaVolterra equations In Example 1(a) we showed that parametric ...
 7.5.23: Modified predatorprey dynamics In Example 1 we used LotkaVolterra...
 7.5.24: Modified aphidladybug dynamics In Exercise 22 we modeled populatio...
Solutions for Chapter 7.5: Phase Plane Analysis
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 7.5: Phase Plane Analysis
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.5: Phase Plane Analysis includes 24 full stepbystep solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 24 problems in chapter 7.5: Phase Plane Analysis have been answered, more than 26073 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.