This

**preview**shows half of the first page. to view the full**2 pages of the document.**Assignment 1

MAT 157

Due: September 19, 2012

1. Consider the set

M={

a b

c d

:a, b, c, d ∈R}

with the usual operations of addition and multiplication:

a b

c d

+

a0b0

c0d0

=

a+a0b+b0

c+c0d+d0

and

a b

c d

·

a0b0

c0d0

=

aa0+bc0ab0+bd0

ca0+dc0cb0+dd0

.

Deﬁne the new operation

[X, Y ] = XY −Y X.

Prove that

(a) For all X, Y ∈ M, we have [X, Y ] = −[Y, X] and in particular, the operation

is not commutative.

(b) There exist elements X, Y, Z ∈ M such that

[[X, Y ], Z]6= [X, [Y, Z]]

and so the operation is not associative.

(c) Prove that for all X, Y, Z ∈ M, we have the identity

[X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0.

2. (Spivak, Chapter 1, Problem 7) Prove that if 0 < a < b, then

a <

√

ab <

1

2

(a+b)< b.

1

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