 9.1: In 14, find a general solution for the system where A is given
 9.2: In 14, find a general solution for the system where A is given
 9.3: In 14, find a general solution for the system where A is given
 9.4: In 14, find a general solution for the system where A is given
 9.5: In 5 and 6, find a fundamental matrix for the system where A is given.
 9.6: In 5 and 6, find a fundamental matrix for the system where A is given.
 9.7: In 710, find a general solution for the system where A and are given.
 9.8: In 710, find a general solution for the system where A and are given.
 9.9: In 710, find a general solution for the system where A and are given.
 9.10: In 710, find a general solution for the system where A and are given.
 9.11: In 11 and 12, solve the given initial value problem
 9.12: In 11 and 12, solve the given initial value problem
 9.13: In 13 and 14, find a general solution for the CauchyEuler system wh...
 9.14: In 13 and 14, find a general solution for the CauchyEuler system wh...
 9.15: In 15 and 16, find the fundamental matrix where A is given
 9.16: In 15 and 16, find the fundamental matrix where A is given
Solutions for Chapter 9: Fundamentals of Differential Equations and Boundary Value Problems 6th Edition
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems  6th Edition
ISBN: 9780321747747
Solutions for Chapter 9
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. Chapter 9 includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6. Since 16 problems in chapter 9 have been answered, more than 2082 students have viewed full stepbystep solutions from this chapter.

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.

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

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

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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.

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

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).

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

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.