- 24.24.1: Compute [3)0 ([4) Ell ) and ([3)0 [4)) Ell ([3)0 ) 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
peA) = det(A - AI) has peA) = zero matrix.
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).
cond(A) = c(A) = IIAIlIIA-III = 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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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+ (Moore-Penrose 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 = Q-1 = Q.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , 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. A-I 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)j-I and det V = product of (Xk - Xi) for k > i.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).