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Comparing corresponding entries in the frst column, we obtain
a
+
b
√
3=
a
−
c
√
3 and
c
+
d
√
a
√
3+
c
, which gives
b
=
−
c
and
d
=
a
. In that case entries in the second
column are automatically equal. We conclude that
B
has the Form
B
=
µ
a
−
c
ca
¶
For arbitrary numbers
a
and
c
.
As in class we conclude that this matrix represents the composition oF a rotation and
a dilation. To see this, it is enough to take
r
=
√
a
2
+
c
2
, and fnd an angle
θ
so that
a
=
r
cos
θ
,
c
=
r
sin
θ
. Then we have:
B
=
µ
r
0
0
r
¶µ
cos
θ
−
sin
θ
sin
θ
cos
θ
¶
.
4. We perForm the algorithm given in class:
⎛
⎝
111

100
320

010
001

⎞
⎠
subtract 3 times row I From row II
⎛
⎝
11 1

0
−
1
−
3
−
310
00 1

⎞
⎠
multiply row II by
−
1
⎛
⎝

013

3
−
10

⎞
⎠
subtract row II From row I
⎛
⎝
−
2
210
01 3

3
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This note was uploaded on 04/12/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.
 Fall '08
 lee

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