Graphics Reference
In-Depth Information
Figure 1.13.
Ordered bases.
(-1,3)
(1,3)
(1,1)
(2,1)
(2,0)
(3,-1)
1
5
2
5
()
=
()
+
()
11
,
13
,
21
,
2
5
6
5
()
=-
()
+
()
20
,
13
,
21
,
and
1
5
2
5
Ê
ˆ
Á
Á
˜
˜
det
=>.
20
2
5
6
5
-
Ë
¯
1.6.3. Example.
To show that the ordered bases ((1,3),(2,1)) and ((3,-1),(-1,3))
determine different orientations of the plane.
Solution.
See Figure 1.13. Note that
(
)
=-
()
+
()
31
,
-
13 221
,
,
7
5
6
5
(
)
=
()
-
()
-
13
,
13
,
21
,
and
-
12
7
5
Ê
Á
ˆ
˜
8
5
det
6
5
=-
<
0.
-
Since arbitrary vector spaces do not have any special bases, one typically cannot
talk about a “standard” orientation, but can only
compare
ordered bases as to
whether they determine the same orientation or not. In the special case of
R
n
we do
have the standard basis (
e
1
,
e
2
,...,
e
n
) though.
