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 1.1.1P: In Problems, verify by substitution that each given function is a s...
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 1.1.12P: ?Prob 12PIn Problems, verify by substitution that each given functi...
 1.1.13P: In Problems, substitute y = erx into the given differential equatio...
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 1.1.27P: In Problems, a function y = g(x) is described by some geometric pro...
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 1.1.43P: (a) If k is a constant, show that a general (oneparameter) solutio...
 1.1.44P: (a) Continuing Problem, assume that k is positive, and then sketch ...
 1.1.45P: Suppose a population P of rodents satisfies the differential equati...
 1.1.46P: Suppose the velocity v of a motorboat coasting in water satisfies t...
 1.1.47P: Example 7 we saw that y(x) = 1/(C ? x) defines a oneparameter fami...
 1.1.48P: (a) Show that y(x) = Cx4 defines a oneparameter family of differen...
Solutions for Chapter 1.1: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 1.1
Get Full SolutionsDifferential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3. Chapter 1.1 includes 49 full stepbystep solutions. Since 49 problems in chapter 1.1 have been answered, more than 11923 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.