 24.24.1: Compute [3)0 ([4) Ell [5]) and ([3)0 [4)) Ell ([3)0 [5]) in Z6. Sho...
 24.24.2: Prove that ([al Ell [b]) 0 [cl = ([al0 [cl) Ell ([bl0 [cD for all [...
 24.24.3: computeImagenM(2'Z).
 24.24.4: Verify that a(b + c) = ab + ac for all a, b, c E M(2, Z). (Here eac...
 24.24.5: Prove that X[ v21 is a ring (Example 24.4).
 24.24.6: Complete the verification that F = M(lR) is a ring (Example 24.5). ...
 24.24.7: Which of the following properties hold in every ring R? What about ...
 24.24.8: Prove that if R is a ring, a, bE R, and ab = ba, then a(b) = (b)a...
 24.24.9: Show that if R is a ring and S is a nonempty set, then the set of a...
 24.24.10: Consider Example 24.5 with/g, as defined there, replaced by / 0 g. ...
 24.24.11: Complete the verification that Example 24.6 is a ring.
 24.24.12: Prove Theorem 24.1. (This can be done by referring to proofs that h...
 24.24.13: Prove that if R is a ring, then each of the following properties ho...
 24.24.14: Prove that a ring has at most one unity.
 24.24.15: LetE denote the set of even integers. Prove that with the usual add...
 24.24.16: Prove that a2  b2 0: (a + b)(a  b) for all a, b in a ring R iff R...
 24.24.17: Prove that (a + b)2 = a2 + 2ab + b2 for all a, b in a ring R iff R ...
 24.24.18: Verify that if A is an Abelian group, with addition as the operatio...
 24.24.19: In the ring of integers, if ab = ae and a i= 0, then b = e. Is this...
 24.24.20: Verify that if R is a ring and a, b ER, then(a + b)3 0: a3 + aba + ...
 24.24.21: Prove that if R is a commutative ring, a, bE R, and n is a positive...
 24.24.22: For each set S, let peS) denote the set of all subsets of S. For A ...
Solutions for Chapter 24: DEFINITION AND EXAMPLES
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 24: DEFINITION AND EXAMPLES
Get Full SolutionsSince 22 problems in chapter 24: DEFINITION AND EXAMPLES have been answered, more than 6348 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 24: DEFINITION AND EXAMPLES includes 22 full stepbystep solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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)·(b  Ax) = o.

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 r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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