 2.2.1P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.2P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.3P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.4P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.5P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.6P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.7P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.8P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.9P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.10P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.11P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.12P: In problem first solve the equation f(x) = 0 to find the critical p...
 2.2.13P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.14P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.15P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.16P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.17P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.18P: In Problem, use a computer system or graphing calculator to plot a ...
 2.2.19P: The differential equation models a logistic population with harvest...
 2.2.20P: The differential equation models a population with stocking at rate...
 2.2.21P: Consider the differential equation dx/dt = kx ? x3 (a) If k 0, show...
 2.2.22P: Consider the differential equation dx/dt = x + kx3 containing the p...
 2.2.23P: Suppose that the logistic equation dx/dt = kx(M ? x) models a popul...
 2.2.24P: Separate variables in the logistic harvesting equation dx/dt = k(N ...
 2.2.25P: Use the alternative forms of the solution in to establish the concl...
 2.2.26P: Example 4 dealt with the case 4h > kM2 in the equation dx/dt = kx(M...
 2.2.27P: Example 4 dealt with the case Ah > kM2 in the equation dx/dt = kx(M...
 2.2.28P: This problem deals with the differential equation dx/dt = kx(x ? M)...
 2.2.29P: Consider the two differential equations each having the critical po...
Solutions for Chapter 2.2: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 2.2
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3. Chapter 2.2 includes 29 full stepbystep solutions. Since 29 problems in chapter 2.2 have been answered, more than 12660 students have viewed full stepbystep solutions from this chapter. Differential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

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