 2.14.1: Find all solutions of each of the following systems of equations.
 2.14.2: Find all solutions of each of the following systems of equations.
 2.14.3: Find all solutions of each of the following systems of equations.
 2.14.4: Find all solutions of each of the following systems of equations.
 2.14.5: Find the solution of each of the following initialvalue problems. ...
 2.14.6: Find the solution of each of the following initialvalue problems. ...
 2.14.7: Find the solution of each of the following initialvalue problems. ...
 2.14.8: Find the solution of each of the following initialvalue problems. ...
 2.14.9: Find the solution of each of the following initialvalue problems. ...
 2.14.10: Find the solution of each of the following initialvalue problems. ...
 2.14.11: Find the solution of each of the following initialvalue problems. ...
 2.14.12: Find the solution of each of the following initialvalue problems. ...
Solutions for Chapter 2.14: The method of elimination for systems
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.14: The method of elimination for systems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Chapter 2.14: The method of elimination for systems includes 12 full stepbystep solutions. Since 12 problems in chapter 2.14: The method of elimination for systems have been answered, more than 5882 students have viewed full stepbystep solutions from this chapter. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.