레만투 REference, MANual, TUtorial

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Matrix Notation

[a b; c d] = | a  b |
                | c  d |

Answers

1.38

exp(ia)exp(ib)
= (sin(a) + icos(a))(sin(b) + icos(b))
= (sin(a)sin(b) - cos(a)cos(b)) + i(sin(a)cos(b) + cos(a)sin(b))

exp(ia)exp(ib)
= exp(i(a+b))
= sin(a+b) + icos(a+b)

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

2.13

When n(S) = n,

∀a ∈ S, n({a*a}) = n
∀a,b ∈ S and a ≠ b, n({a*b}) = n2 - n

* is commutative, so the number of cases is (n2 - n)/2 + n = (n2 +n) / 2

nΠ((n2 +n) / 2) = n^((n2 +n) / 2)

2.37

Let c = a*b ( a, b ∈ H )

c*c = (a*b)*(a*b)
     = (a*a)*(b*b)
     = a*b
     = c

∴ The set H is closed under *.

3.25 Continuing the ideas of Excercise 24 can a binary structure have a left identity element e_L and a right identity element e_R where e_L ≠ e_R? If so, give an example, using an operation on a finite set S. If not, prove that it is impossible.

Let eL is a left identity element, and eR is a right identity element.

eL*eR = eR (because eL*a=a)
eL*eR = eL (because a*eR=a)

∴ eL = eL*eR = eR

3.33 Let H be the subset of M₂(R) consisting of all matrices of the form [a -b; b a] for a, b ∈ R. Exercise 23 of Section 2 shows that H is closed under both matrix addition and matrix multiplication.

a. Show that <C, +> is isomorphic to <H, +>

Let a + bi = [a -b; b a],

(a + bi) + (c + di) 
= (a+c) + (b+d)i
= [(a+c) -(b+d); (b+d) (a+c)]
= [a -b; b a] + [c -d; d c] 

b. Show that <C, ·> is isomorphic to <H, ·>

In the same way,

(a + bi)·(c + di) 
= (ac-bd) + (ad+bc)i
= [(ac-bd) -(ad+bc); (ad+bc) (ac-bd)]
= [a -b; b a][c -d; d c]

(We say that H is a matrix representation of the complex numbers C)

3.34 There are 16 possible binary structure on the set {a,b} of two elements. How many nonisomorphic (that is, structurally different) structures are there among 16?

(a*a, a*b, b*a, b*b)s are

(a,a,a,a), (a,a,a,b), (a,a,b,a), (a,a,b,b), 
(a,b,a,a), (a,b,a,b), (a,b,b,a),(a,b,b,b),
(b,a,a,a), (b,a,a,b), (b,a,b,a), (b,a,b,b), 
(b,b,a,a), (b,b,a,b), (b,b,b,a),(b,b,b,b)

There are only one isomorphism, a ↔ b. (a*a, a*b, b*a, b*b) ↔ (b*'b, b*'a, a*'b, a*'a). So, interchange a and b and reverse the listed order.

(a,a,a,a) ~ (b,b,b,b)
(a,a,a,b) ~ (a,b,b,b)
(a,a,b,a) ~ (b,a,b,b)
(a,a,b,b) ~ (b,b,a,a)
(a,b,a,a) ~ (b,b,a,b)
(a,b,a,b)
(a,b,b,a) ~ (b,a,a,b)
(b,a,a,a), ~ (b,b,b,a)
(b,a,b,a)

∴ There are 9 isomorphic strurctures.