 10.2.1: 14 Show that x1std and x2std are solutions to the system of differe...
 10.2.2: 14 Show that x1std and x2std are solutions to the system of differe...
 10.2.3: 14 Show that x1std and x2std are solutions to the system of differe...
 10.2.4: 14 Show that x1std and x2std are solutions to the system of differe...
 10.2.5: 58 Show that x1std and x2std are solutions to the initialvalue pro...
 10.2.6: 58 Show that x1std and x2std are solutions to the initialvalue pro...
 10.2.7: 58 Show that x1std and x2std are solutions to the initialvalue pro...
 10.2.8: 58 Show that x1std and x2std are solutions to the initialvalue pro...
 10.2.9: Prove the Superposition Principle.
 10.2.10: Show that if the eigenvalues of a 2 X 2 matrix are real and distinc...
 10.2.11: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.12: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.13: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.14: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.15: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.16: 1116 Sketch several solution curves in the phase plane of the syste...
 10.2.17: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c2321...
 10.2.18: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c12 2...
 10.2.19: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c1 04...
 10.2.20: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c21 2...
 10.2.21: 1728 Solve the initial value problem dxydt Ax with xs0d x0. c23 426...
 10.2.22: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c 0 1...
 10.2.23: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c21 2...
 10.2.24: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c3 00...
 10.2.25: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c 0 2...
 10.2.26: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c4 22...
 10.2.27: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c2 25...
 10.2.28: 1728 Solve the initial value problem dxydt Ax with xs0d x0. A c3 24...
 10.2.29: In Exercise 10.1.24 we considered the nongeneric system of differen...
 10.2.30: When the eigenvalues of the coefficient matrix are complex, the ori...
 10.2.31: Our focus has been on systems whose coefficient matrices have disti...
 10.2.32: A slightly more complicated system with repeated eigenvalues is dx ...
 10.2.33: 3335 The system of differential equations dxydt Ax depends on a rea...
 10.2.34: 3335 The system of differential equations dxydt Ax depends on a rea...
 10.2.35: 3335 The system of differential equations dxydt Ax depends on a rea...
 10.2.36: Use the general solution (Equation 8) and Eulers formula to prove T...
 10.2.37: Use Theorem 15 to prove the trace and determinant condition for sta...
 10.2.38: Justify the summary of the qualitative behavior depicted in Figure ...
 10.2.39: Provide an argument, based on Theorem 2, for why solution curves in...
Solutions for Chapter 10.2: Qualitative Analysis of Linear Systems
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 10.2: Qualitative Analysis of Linear Systems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 39 problems in chapter 10.2: Qualitative Analysis of Linear Systems have been answered, more than 27336 students have viewed full stepbystep solutions from this chapter. Chapter 10.2: Qualitative Analysis of Linear Systems includes 39 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column space C (A) =
space of all combinations of the columns of A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
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.

GramSchmidt 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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.