 1.17.1: In each of 15, find the solution of the given initialvalue proble...
 1.17.2: In each of 15, find the solution of the given initialvalue proble...
 1.17.3: In each of 15, find the solution of the given initialvalue proble...
 1.17.4: In each of 15, find the solution of the given initialvalue proble...
 1.17.5: In each of 15, find the solution of the given initialvalue proble...
Solutions for Chapter 1.17: What to do in practice
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 1.17: What to do in practice
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.17: What to do in practice includes 5 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. Since 5 problems in chapter 1.17: What to do in practice have been answered, more than 6125 students have viewed full stepbystep solutions from this chapter.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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

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

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.

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.

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.

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.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.