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9. (d) Recall that the symbol Qt represents all positive rationals. Is the group (Z,...
* (9) Let n be a positive integer. Define : Z → Zn by (k) =...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
Qi Consider the group (D x Q, +), where Q Q = {(a,b)|a, b E Q},...
Qi Consider the group (D x Q, +), where Q Q = {(a,b)|a, b E Q}, and where addition is defined in the usual way by (a, b) +(c,d) = (a +c, b+d). So, for instance, (,-) € QxQ, and (2, - ) + (1, 1) = (1, 1). (a) What is the identity in this group? You do not need to justify your answer. (b) What is the inverse of the element (x, y) E Q? You do not...
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined...
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3 is not an isomorphism.
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3...
(10) Recall that for a unit vector ū= in R2, the matrix P = ūūt represents...
(10) Recall that for a unit vector ū= in R2, the matrix P = ūūt represents the projection on ū. (a) Are there values a and b such that P is a SPD matrix? Explain. (b) Orthogonally diagonalize P. (c) Orthogonally diagonalize the reflection matrix L = 2P - I. (10) Determine the range of a so that the quadratic form Q(2, y, z) = a(z?+y2 +22)+2xy-2yz+2zz is positive definite.
e=[(0,1)] Recall that a rational number x = [(a,b)] is positive if ab > 0 in...
e=[(0,1)]
Recall that a rational number x = [(a,b)] is positive if ab > 0 in Z and negative if ab < 0 in Z (for any choice of representative (a,b) E x). For x,y EQ we say x <y iff y e(-x) is positive. (a) Show that x <y iff x (-y) is negative. (b) Show that for each x E Q, precisely one of the following three statements is true: x = e, e < x, x<e. (c)...
7. Let z x+y (a) Show that f(z) z3 is analytic. 4 marks Recall the Caucy-Riemann equations are: ди ди an d_ where f (z)...
7. Let z x+y (a) Show that f(z) z3 is analytic. 4 marks Recall the Caucy-Riemann equations are: ди ди an d_ where f (z) -u(x, y) + iv(x, y). (b) Let x2 and y 1 such that z-2i is a solution to 2abi [3 marks] Determine a and b (c) Find all other solutions of 23-a + bi in polar form correct to 2 significant 3 marks] figures If you were not able to solve for a and b...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition se...
just part c and d
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition series. Explain why your example works. (b) Find a refinement of the series you gave above which is a composition series. That is, show how to extend your series into a composition series. (c) Find all other composition series and briefly explain how you know you have them all. (d) If p(x) is a polynomial whose splitting...
Numbers 3,4,11 a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real...
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
m) Recall z = { 07, s... al. (200) anabelian group and (2010) isa Semigroup with...
m) Recall z = { 07, s... al. (200) anabelian group and (2010) isa Semigroup with identity [1] Let A- 41],3], [-],[9]} B-{{], [2], [4][5][6], and oil is a commutative Ist a binary operation on A? Give reasons it is, write out its multiplication table. 2) is a binary operation on B? Give Resisons! If it is, write out its multiplication table. 3 25 is a binary operation on A? f it is, write out its multiplication Give reasons. table....
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
Qi Consider the group (D x Q, +), where Q Q = {(a,b)|a, b E Q}, and where addition is defined in the usual way by (a, b) +(c,d) = (a +c, b+d). So, for instance, (,-) € QxQ, and (2, - ) + (1, 1) = (1, 1). (a) What is the identity in this group? You do not need to justify your answer. (b) What is the inverse of the element (x, y) E Q? You do not...
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3 is not an isomorphism.
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3...
(10) Recall that for a unit vector ū= in R2, the matrix P = ūūt represents the projection on ū. (a) Are there values a and b such that P is a SPD matrix? Explain. (b) Orthogonally diagonalize P. (c) Orthogonally diagonalize the reflection matrix L = 2P - I. (10) Determine the range of a so that the quadratic form Q(2, y, z) = a(z?+y2 +22)+2xy-2yz+2zz is positive definite.
e=[(0,1)]
Recall that a rational number x = [(a,b)] is positive if ab > 0 in Z and negative if ab < 0 in Z (for any choice of representative (a,b) E x). For x,y EQ we say x <y iff y e(-x) is positive. (a) Show that x <y iff x (-y) is negative. (b) Show that for each x E Q, precisely one of the following three statements is true: x = e, e < x, x<e. (c)...
7. Let z x+y (a) Show that f(z) z3 is analytic. 4 marks Recall the Caucy-Riemann equations are: ди ди an d_ where f (z) -u(x, y) + iv(x, y). (b) Let x2 and y 1 such that z-2i is a solution to 2abi [3 marks] Determine a and b (c) Find all other solutions of 23-a + bi in polar form correct to 2 significant 3 marks] figures If you were not able to solve for a and b...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
just part c and d
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition series. Explain why your example works. (b) Find a refinement of the series you gave above which is a composition series. That is, show how to extend your series into a composition series. (c) Find all other composition series and briefly explain how you know you have them all. (d) If p(x) is a polynomial whose splitting...
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
m) Recall z = { 07, s... al. (200) anabelian group and (2010) isa Semigroup with identity [1] Let A- 41],3], [-],[9]} B-{{], [2], [4][5][6], and oil is a commutative Ist a binary operation on A? Give reasons it is, write out its multiplication table. 2) is a binary operation on B? Give Resisons! If it is, write out its multiplication table. 3 25 is a binary operation on A? f it is, write out its multiplication Give reasons. table....
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