Math 361
Final Exam (Spring 2015)
I. Roots of Unity
1. Write the group of roots of unity for z5 = 1, expressing
each complex element in polar form. Show your
calculations.
3. Indicate by arrows and two lists of elements an
isomorphism between the group you found in problem 1
and a specific modulo number group.
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2. List pairs of inverses in the group you found in
problem 1.
4a) Prove or disprove: the group of roots of unity for z5 =
1 is a subgroup of the group of roots of unity for the
complex equation z12 = 1.
b) Name the smallest group of which the group of roots
of unity for z5 = 1 is a subgroup. (No proof needed)
II. Complex Mappings
Consider the function w = (1-z)/(1+z). Graph the function by denoting which relatively nice domain subsets are
transformed into relatively nice co-domain subsets. Hint: The imaginary axis is wrapped around the origin. (Show
this with examples) What subsets maps inside and outside of that circle?
Domain
Codomain
Succinct Verbal Description:
Math 361
Final Exam (Spring 2015)
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III. Euclidean Geometry and the Group of Transformations.
1. Prove that the arbitrary transformation
2. Write in proper form the inverse of a transformation
(𝑇(𝑎) , 𝑂) has an inverse, where O is any element of
𝑏
𝑎
O(2,R), and vector ( ) is any element of 𝑅2.
𝑏
IV. Spherical Geometry and the Group of transformations.
1. O(3,R) is the set of all 3x3 orthogonal matrices, each of
which is either a rotation, a reflection, or an improper
rotation (rotation and reflection). The definition of the
group is the same as for O(2,R). That is,
O ∈ O(3,R) if O*OT = I.
Prove the following rotation about the z-axis is
orthogonal:
𝑐𝑜𝑠∅ −𝑠𝑖𝑛∅ 0
Oφ,z = [ 𝑠𝑖𝑛∅ 𝑐𝑜𝑠∅ 0]
0
0
1
(𝑇(𝑎) , 𝑂) that reflects vectors about y = -x and then
𝑏
translates them 2 units up 3 units left. Show work neatly.
2. State the inverse of matrix Oφ,z in problem 1 and state
𝑥
its action on an arbitrary 3-D vector (𝑦) when ∅ > 0.
𝑧
Draw a 3-D diagram with x,y, and z axes to make your
statement clear.
Math 361
Final Exam (Spring 2015)
3. Write two additional rotational matrices in O(3,R):
a) Matrix Oψ,x which rotates in 3-D about the x-axis.
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4. State 3 orthogonal matrices that reflect 3-D vectors
a) across the x-y plane (mirror);
b) across the y-z plane (mirror);
b) Matrix Oτ,y which rotates in 3-D about the y-axis.
c) across the x-z plane (mirror).
𝑐𝑜𝑠∅ −𝑠𝑖𝑛∅ 0
5. Prove that Oφ,z = [ 𝑠𝑖𝑛∅ 𝑐𝑜𝑠∅ 0] preserves 3-D Euclidean distance.
0
0
1
Math 361
Final Exam (Spring 2015)
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IV. Prove or disprove: the set of all complex numbers z such that |z| = 1 is an Abelian group under complex
multiplication. Hint: Let C ={eiϴ| ϴ ∈ R}.
Math 361
Final Exam (Spring 2015)
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V. Ring Homomorphisms
a) Is there a homomorphism from zmod12 into zmod3? If so, indicate the homomorphism by listing where each
element of zmod12 is mapped. Also state the IDEAL induced in the mapping! If the homomorphism does not exist,
show why not.
b) The function y = f(x) = 10x is defined on all real numbers. Explain with examples how f and its inverse function f-1 are
isomorphisms of groups, using the scaffold below.
1. Precisely define the isomorphic groups in this case.
2. Show how f and f-1 take group identities to group
identities.
3. By way of examples, show the 3 laws of exponents are
really homomorphism properties (preserving structure of
respective groups).
4. By way of examples, show the 3 laws of logarithms are
really homomorphism properties.
Math 361
Final Exam (Spring 2015)
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