MAT 312/AMS 351 Fall 2010 Homework 8 1. In p. 56 problem (6), check that the rst block (13615) decodes to 40 = O. You may do this as follows: rst check that the multiplicative inverse of 121 modulo (23711) = 23400 is 3481. Show your work. Then use a calcu
MAT 312/AMS 351 Fall 2010 Review for Midterm 2 1.7. Understand that if (a, n) = 1, then the equation ax b mod n has a unique solution (mod n) and know how to nd it. That is the simplest case. Example 1.68 p.45. More generally, understand that if (a, n) =
MAT 312/AMS 351 Notes and exercises on normal subgroups and quotient groups. If H is a subgroup of G, the equivalence relation H is dened between elements of G as follows: g1 H g2 h H, g1 = g2 h. Proposition 1. This is indeed an equivalence relation. Proo
MAT 312/AMS 351 Notes and Exercises on Permutations and Matrices. We can represent a permutation S (n) by a matrix M in the following useful way. If (i) = j , then M has a 1 in column i and row j ; the entries are 0 otherwise. This M permutes the unit col
MAT 312/AMS 351 Fall 2010 Homework 11 1. A linear fractional transformation f (x) is a function of the form ax + b f ( x) = cx + d where a, b, c, d, are real numbers satisfying ad bc = 1. Show that if f (x) as above and g (x) = (ex + f )/(gx + h) are line
MAT 312/AMS 351 Fall 2010 Homework 6 1. Find x satisfying x 2 mod 7 x 9 mod 23 2. Find x satisfying
x 5 mod 11
3. page 54 problem (1)
x 3 mod 5 x 4 mod 7
4. page 54 problem (3). Hint: Show that a a5 mod 5 and mod 2.
1
MAT 312/AMS 351 Fall 2010 Homework 5 1. Explain in your own words why, if n is a prime, a linear congruence equation ax n b always has a solution (i.e that given any integers a and b, there exists an integer x such that ax b is a multiple of n), and that
MAT 312/AMS 351 Fall 2010 Homework 4 1. p. 43 Exercise 1. 2. p. 43 Exercise 3. Consider also the numbers 9 and 11. 3. Given positive integers a and b, suppose there exist integers k and such that 1 = ka + b. Show that a and b must be relatively prime. 4.
Stony Broo University MAT 312/AMS 351 Fall 2010 Notes on Hamming Codes 1. Error-detecting matrices. Suppose a group code is generated by a matrix
G=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 0 0
0 1 1 0
0 1 1 1
We can write this matrix as G = (I4 |A)
.
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