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Change of Basis in 2D

​
old basis
new basis
x
3.000
5.000
=
(
1
v
,
2
v
)
1.500
-1.500
0.500
1.300
z
4.222
2.222
det (
1
v
,
2
v
) =
2.700
Given a point
P
in the plane you can see its coordinates
(
x
1
,
x
2
)
with respect to the standard basis
(
1
e
,
2
e
)
, or compute its coordinates
(
z
1
,
z
2
)
with respect to a different basis
(
1
v
,
2
v
)
. The signed area of the parallelogram determined by this basis,
det(
1
v
,
2
v
)
, is a crucial quantity.
This Demonstration complements the Demonstration "Coordinates of a Point Relative to a Basis in 2D", by E. Schulz, which shows how a point varies as its coordinates change.
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