 41.41.1: Prove Theorem 41.1.
 41.41.2: Assume that D is an integral domain and that a, bED.(a) Prove that ...
 41.41.3: If a and b are elements of a principal ideal domain D, not both zer...
 41.41.4: Prove that if a and b are elements of a principal ideal domain D wi...
 41.41.5: Prove that if a, b, andp are nonzero elements ofa principal ideal d...
 41.41.6: Prove that if D is a principal ideal domain. ai, a2, ... , a", p E ...
 41.41.7: If D is a principal ideal domain, a E D, and a is not a unit of D, ...
 41.41.8: Use 41.6 and 41.7 to prove Theorem 41.2.
 41.41.9: Consider the four classes of rings in (41.1). Determine the smalles...
 41.41.10: (a) Which integers k E {I, 2, ... , to} are quadratic residues of I...
 41.41.11: Prove that if m and n are positive integers with m > n, and x = m2 ...
 41.41.12: Prove that a nonzero ideal (a) of a principal ideal domain D is a m...
Solutions for Chapter 41: FACTORIZATION AND IDEALS
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 41: FACTORIZATION AND IDEALS
Get Full SolutionsSince 12 problems in chapter 41: FACTORIZATION AND IDEALS have been answered, more than 8139 students have viewed full stepbystep solutions from this chapter. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 41: FACTORIZATION AND IDEALS includes 12 full stepbystep solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6.

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.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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).

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).