 3.3.1: The system in Exercise 1, Section 3.2
 3.3.2: The system in Exercise 2, Section 3.2
 3.3.3: The system in Exercise 3, Section 3.2
 3.3.4: The system in Exercise 6, Section 3.2
 3.3.5: The system in Exercise 7, Section 3.2
 3.3.6: The system in Exercise 8, Section 3.2
 3.3.7: The system in Exercise 9, Section 3.2
 3.3.8: The system in Exercise 10, Section 3.2
 3.3.9: The initialvalue problems in Exercise 11, Section 3.2
 3.3.10: The initialvalue problems in Exercise 12, Section 3.2
 3.3.11: The initialvalue problems in Exercise 13, Section 3.2
 3.3.12: The initialvalue problems in Exercise 14, Section 3.2
 3.3.13: The secondorder equation in Exercise 21, Section 3.2
 3.3.14: The secondorder equation in Exercise 22, Section 3.2
 3.3.15: The secondorder equation in Exercise 23, Section 3.2
 3.3.16: The secondorder equation in Exercise 24, Section 3.2
 3.3.17: dx dt dy dt = 2 1 0 1 x y 18
 3.3.18: dx dt dy dt = 2 1 1 1 x y 19.
 3.3.19: The slope field for the system dx dt = 2x + 1 2 y dy dt = y is show...
 3.3.20: The slope field for the system dx dt = 2x + 6y dy dt = 2x 2y is sho...
 3.3.21: For the harmonic oscillator with mass m = 1, spring constant k = 6,...
 3.3.22: Consider a harmonic oscillator with mass m = 1, spring constant k =...
 3.3.23: dY dt = 0.2 0.1 0.0 0.1 Y
 3.3.24: dY dt = 0.1 0.2 0.0 1.0 Y 2
 3.3.25: dY dt = 0.2 0.1 0.0 0.1 Y 2
 3.3.26: dY dt = 0.1 0.0 0.2 0.2 Y 2
 3.3.27: Consider the linear system dY dt = 2 1 0 2 Y. (a) Show that (0, 0) ...
Solutions for Chapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES
Get Full SolutionsChapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES includes 27 full stepbystep solutions. Since 27 problems in chapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES have been answered, more than 15662 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. This expansive textbook survival guide covers the following chapters and their solutions.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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

Kirchhoff's Laws.
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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.