Prove that the set of all 3 × 3 matrices with real entries of the form [ 1 a b 0 1 c 0 0 1 ] is a group. (Multiplication is defined by [ 1 a b 0 1 c 0 0 1 ] [ 1 a ' b ' 0 1 c ' 0 0 1 ] = [ 1 a + a ' b ' + a c ' + b 0 1 c ' + c 0 0 1 ] . This group, sometimes called the Heisenberg group after the NobelPrize−winning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)
Prove that the set of all 3 × 3 matrices with real entries of the form [ 1 a b 0 1 c 0 0 1 ] is a group. (Multiplication is defined by [ 1 a b 0 1 c 0 0 1 ] [ 1 a ' b ' 0 1 c ' 0 0 1 ] = [ 1 a + a ' b ' + a c ' + b 0 1 c ' + c 0 0 1 ] . This group, sometimes called the Heisenberg group after the NobelPrize−winning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)
Solution Summary: The author explains that G is said to be group under the operation ast , if it satisfies all the following properties.
Prove that the set of all
3
×
3
matrices with real entries of the form
[
1
a
b
0
1
c
0
0
1
]
is a group. (Multiplication is defined by
[
1
a
b
0
1
c
0
0
1
]
[
1
a
'
b
'
0
1
c
'
0
0
1
]
=
[
1
a
+
a
'
b
'
+
a
c
'
+
b
0
1
c
'
+
c
0
0
1
]
.
This group, sometimes called the Heisenberg group after the NobelPrize−winning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)
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