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In short, we can write
µ
x
y
¶
A
7→
µ
x
0
y
0
¶
B
7→
µ
x
0
y
0
¶
.
Since the two lines are perpendicular, we see from the picture that these 3 points are
vertices of a rightangled triangle and that the origin is at the midpoint of its hypotenuse.
Thus
µ
x
0
y
0
¶
=
−
µ
x
y
¶
=
µ
−
x
−
y
¶
.
Since the matrix product
BA
corresponds to the composition of
A
followed by
B
,we
conclude
µ
x
y
¶
=
µ
−
x
−
y
¶
,
so
is the reﬂection about the origin, i.e. the rotation by 180
◦
. Now we can write the
matrix:
=
µ
cos(180
◦
)
−
sin(180
◦
)
sin(180
◦
)
cos(180
◦
)
¶
=
µ
−
10
0
−
1
¶
.
This can also be seen from
µ
x
y
¶
=
µ
−
x
−
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 Fall '08
 lee

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