- Chapter 8.1: For real numbers x and y, define an operation x y by x y D p x 2 C ...
- Chapter 8.2: In Exercise 40.2 we considered the operation x ? y D x C y xy for r...
- Chapter 8.3: In you were asked to show that .R 0 ; ?/ is an Abelian group where ...
- Chapter 8.4: List the elements in Z 32 and find '.32/.
- Chapter 8.5: Consider the group .Z 15; /. Find the following subsets of Z 15: a....
- Chapter 8.6: Let .G; / be an Abelian group. Define the following two subsets of ...
- Chapter 8.7: Let .G; / be a group with exactly three elements. Prove that G is i...
- Chapter 8.8: Find an isomorphism between .Z 13; / and .Z12; /.
- Chapter 8.9: . Let .G; / be a group and let .H; / and .K; / be subgroups. Define...
- Chapter 8.10: Show that for all elements g of .Z 15; /, we have g 4 D 1. Use this...
- Chapter 8.11: Without the use of any computational aid, calculate 2 90 mod 89.
- Chapter 8.12: Let n D 38168467. Use the fact that 2 n 6178104 .mod n/ to determin...
- Chapter 8.13: Let n D 38168467. Given that '.n/ D 38155320, calculate (without th...
- Chapter 8.14: Using only a basic hand-held calculator, compute 874256 mod 9432:
- Chapter 8.15: Find all values of p 71 in Z883.
- Chapter 8.16: Find all values of p 1 in Z440617. Note that 440617 factored into p...
- Chapter 8.17: Let n D 5460947. In Zn we have 12359072 D 18424122 D 36185352 D 422...
- Chapter 8.18: Alice and Bob communicate using the Rabin public-key cryptosystem. ...
- Chapter 8.19: Alice and Bob switch to using the RSA public-key cryptosystem. Alic...
- Chapter 8.20: Bob sends Alice a message using Alices RSA public key (as described...
- Chapter 8.21: Given that n D 40119451 is the product of two distinct primes and '...
Solutions for Chapter Chapter 8: Algebra
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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).
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Every v in V is orthogonal to every w in W.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).