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4. Find a subgroup of GL(3,R) that is isomorphic to 53. Hint consider what happens when...
What is GL(3,R) / SL(3,R) isomorphic to?
What is GL(3,R) / SL(3,R) isomorphic to?
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with w...
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answ...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the ...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
Prove that Z/ ≡3 has exactly three elements using the given hint! Definition: Let R be...
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
How many non-isomorphic unital rings are there of order 4? Question 3: How many non-isomorphic unital...
How many non-isomorphic unital rings are there of order 4?
Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the a...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
Q4 only: Question 3. Consider the region of R3 given by V is bounded by three surfaces. Si is a disc of radius 1 in the plane z -0. S3 is a disc of radius 2 in the plane z 3 and a) Make a clear sketc...
Q4 only:
Question 3. Consider the region of R3 given by V is bounded by three surfaces. Si is a disc of radius 1 in the plane z -0. S3 is a disc of radius 2 in the plane z 3 and a) Make a clear sketch of V. (Hint: You could consider the cross-section of S2 with y-0, and then use the circular symmetry. (b) Express V in cylindrical coordinates. (c) Calculate the volume of V, working in cylindrical...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 +...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
3. What happens when an excise tax is paid mainly by consumers? 4. Describe what happens...
3. What happens when an excise tax is paid mainly by consumers? 4. Describe what happens when an excise tax is paid mainly by producers?
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
How many non-isomorphic unital rings are there of order 4?
Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
Q4 only:
Question 3. Consider the region of R3 given by V is bounded by three surfaces. Si is a disc of radius 1 in the plane z -0. S3 is a disc of radius 2 in the plane z 3 and a) Make a clear sketch of V. (Hint: You could consider the cross-section of S2 with y-0, and then use the circular symmetry. (b) Express V in cylindrical coordinates. (c) Calculate the volume of V, working in cylindrical...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
3. What happens when an excise tax is paid mainly by consumers? 4. Describe what happens when an excise tax is paid mainly by producers?
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