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HOMEWORK 2

Due Thursday, April 20, at the beginning of discussion

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1. Let n 2 N. Find all complex solutions to the equation: zn = i. 2. In this problem, you will prove the sin and cos sum formulas in two ways.

sin(a + b) = sin a cos b + cos a sin b

cos(a + b) = cos a cos b � sin a sin b • Use Euler’s formula: eia = cos a + i sin a to prove the formulas. • Use the rotation matrix R✓ from last term and the fact that the matrix of a com- position of two linear transformations is the product of their respective matrices.

(therefore R✓R⌫ = R✓+⌫)

3. Prove Euler’s formula eia = cos a+i sin a by using the power series expansion of ex, sin x

and cos x.

4. Fill the table for the binary operation in a way that makes the operation associative.

5. Let M2(R) be the set of 2 ⇥ 2 matrices with real entries, and let K be the subset of M2(R) defined by

K =

( a b

�b a

! : a, b 2 R

) .

• Show that addition of matrices is a binary operation on K. • Is (K, +) a group? Prove your answer. • Show that (K, +) is isomorphic to (C, +). (You need to build a map that associates a complex number to each matrix in K, and you mush show that your map is an

isomorphism.)

• Show that multiplication of matrices is a binary operation on K. • Is (K, ⇤) a group? Prove your answer. • Show that (K, ·) is isomorphic to (C, ·).

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6. Let U be a set and let X be the power set of U (that is, the set of all subsets of U).

Consider the operation of symmetric di↵erence of sets, defined by

A4B = (A [ B) � (A \ B) = (A � B) [ (B � A).

The operation of symmetric di↵erence is a binary operation on X.

a) Show that 4 is commutative. b) Is there an identity element?

c) Does every set A have an inverse? What is it?

d) Is (U, 4) a group?

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