- 8.1: For real numbers x and y, define an operation x y by x y D p x 2 C ...
- 8.2: In Exercise 40.2 we considered the operation x ? y D x C y xy for r...
- 8.3: In you were asked to show that .R 0 ; ?/ is an Abelian group where ...
- 8.4: List the elements in Z 32 and find '.32/.
- 8.5: Consider the group .Z 15; /. Find the following subsets of Z 15: a....
- 8.6: Let .G; / be an Abelian group. Define the following two subsets of ...
- 8.7: Let .G; / be a group with exactly three elements. Prove that G is i...
- 8.8: Find an isomorphism between .Z 13; / and .Z12; /.
- 8.9: . Let .G; / be a group and let .H; / and .K; / be subgroups. Define...
- 8.10: Show that for all elements g of .Z 15; /, we have g 4 D 1. Use this...
- 8.11: Without the use of any computational aid, calculate 2 90 mod 89.
- 8.12: Let n D 38168467. Use the fact that 2 n 6178104 .mod n/ to determin...
- 8.13: Let n D 38168467. Given that '.n/ D 38155320, calculate (without th...
- 8.14: Using only a basic hand-held calculator, compute 874256 mod 9432:
- 8.15: Find all values of p 71 in Z883.
- 8.16: Find all values of p 1 in Z440617. Note that 440617 factored into p...
- 8.17: Let n D 5460947. In Zn we have 12359072 D 18424122 D 36185352 D 422...
- 8.18: Alice and Bob communicate using the Rabin public-key cryptosystem. ...
- 8.19: Alice and Bob switch to using the RSA public-key cryptosystem. Alic...
- 8.20: Bob sends Alice a message using Alices RSA public key (as described...
- 8.21: Given that n D 40119451 is the product of two distinct primes and '...
Solutions for Chapter 8: Algebra
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Tv = Av + Vo = linear transformation plus shift.
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.)
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
A sequence of steps intended to approach the desired solution.
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.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
Outer product uv T
= column times row = rank one matrix.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.