12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Active and passive transformation | 2/2 | https://en.wikipedia.org/wiki/Active_and_passive_transformation | reference | science, encyclopedia | 2026-05-05T13:41:22.578367+00:00 | kb-cron |
The distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces. Fix a finite-dimensional vector space
V
{\displaystyle V}
over a field
K
{\displaystyle K}
(thought of as
R
{\displaystyle \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
), and a basis
B
=
{
e
i
}
1
≤
i
≤
n
{\displaystyle {\mathcal {B}}=\{e_{i}\}_{1\leq i\leq n}}
of
V
{\displaystyle V}
. This basis provides an isomorphism
C
:
K
n
→
V
{\displaystyle C:K^{n}\rightarrow V}
via the component map
(
v
i
)
1
≤
i
≤
n
=
(
v
1
,
⋯
,
v
n
)
↦
∑
i
v
i
e
i
{\textstyle (v_{i})_{1\leq i\leq n}=(v_{1},\cdots ,v_{n})\mapsto \sum _{i}v_{i}e_{i}}
. An active transformation is then an endomorphism on
V
{\displaystyle V}
, that is, a linear map from
V
{\displaystyle V}
to itself. Taking such a transformation
τ
∈
End
(
V
)
{\displaystyle \tau \in {\text{End}}(V)}
, a vector
v
∈
V
{\displaystyle v\in V}
transforms as
v
↦
τ
v
{\displaystyle v\mapsto \tau v}
. The components of
τ
{\displaystyle \tau }
with respect to the basis
B
{\displaystyle {\mathcal {B}}}
are defined via the equation
τ
e
i
=
∑
j
τ
j
i
e
j
{\textstyle \tau e_{i}=\sum _{j}\tau _{ji}e_{j}}
. Then, the components of
v
{\displaystyle v}
transform as
v
i
↦
τ
i
j
v
j
{\displaystyle v_{i}\mapsto \tau _{ij}v_{j}}
. A passive transformation is instead an endomorphism on
K
n
{\displaystyle K^{n}}
. This is applied to the components:
v
i
↦
T
i
j
v
j
=:
v
i
′
{\displaystyle v_{i}\mapsto T_{ij}v_{j}=:v'_{i}}
. Provided that
T
{\displaystyle T}
is invertible, the new basis
B
′
=
{
e
i
′
}
{\displaystyle {\mathcal {B}}'=\{e'_{i}\}}
is determined by asking that
v
i
e
i
=
v
i
′
e
i
′
{\displaystyle v_{i}e_{i}=v'_{i}e'_{i}}
, from which the expression
e
i
′
=
(
T
−
1
)
j
i
e
j
{\displaystyle e'_{i}=(T^{-1})_{ji}e_{j}}
can be derived. Although the spaces
End
(
V
)
{\displaystyle {\text{End}}(V)}
and
End
(
K
n
)
{\displaystyle {\text{End}}({K^{n}})}
are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis
B
{\displaystyle {\mathcal {B}}}
allows construction of an isomorphism.
=== As left- and right-actions === Often one restricts to the case where the maps are invertible, so that active transformations are the general linear group
GL
(
V
)
{\displaystyle {\text{GL}}(V)}
of transformations while passive transformations are the group
GL
(
n
,
K
)
{\displaystyle {\text{GL}}(n,K)}
. The transformations can then be understood as acting on the space of bases for
V
{\displaystyle V}
. An active transformation
τ
∈
GL
(
V
)
{\displaystyle \tau \in {\text{GL}}(V)}
sends the basis
{
e
i
}
↦
{
τ
e
i
}
{\displaystyle \{e_{i}\}\mapsto \{\tau e_{i}\}}
. Meanwhile a passive transformation
T
∈
GL
(
n
,
K
)
{\displaystyle T\in {\text{GL}}(n,K)}
sends the basis
{
e
i
}
↦
{
∑
j
(
T
−
1
)
j
i
e
j
}
{\textstyle \{e_{i}\}\mapsto \left\{\sum _{j}(T^{-1})_{ji}e_{j}\right\}}
. The inverse in the passive transformation ensures the components transform identically under
τ
{\displaystyle \tau }
and
T
{\displaystyle T}
. This then gives a sharp distinction between active and passive transformations: active transformations act from the left on bases, while the passive transformations act from the right, due to the inverse. This observation is made more natural by viewing bases
B
{\displaystyle {\mathcal {B}}}
as a choice of isomorphism
Φ
B
:
K
n
→
V
{\displaystyle \Phi _{\mathcal {B}}:K^{n}\rightarrow V}
. The space of bases is equivalently the space of such isomorphisms, denoted
Iso
(
K
n
,
V
)
{\displaystyle {\text{Iso}}(K^{n},V)}
. Active transformations, identified with
GL
(
V
)
{\displaystyle {\text{GL}}(V)}
, act on
Iso
(
K
n
,
V
)
{\displaystyle {\text{Iso}}(K^{n},V)}
from the left by composition, that is if
τ
{\displaystyle \tau }
represents an active transformation, we have
Φ
B
′
=
τ
∘
Φ
B
{\displaystyle \Phi _{\mathcal {B'}}=\tau \circ \Phi _{\mathcal {B}}}
. On the opposite, passive transformations, identified with
GL
(
n
,
K
)
{\displaystyle {\text{GL}}(n,K)}
acts on
Iso
(
K
n
,
V
)
{\displaystyle {\text{Iso}}(K^{n},V)}
from the right by pre-composition, that is if
T
{\displaystyle T}
represents a passive transformation, we have
Φ
B
″
=
Φ
B
∘
T
{\displaystyle \Phi _{\mathcal {B''}}=\Phi _{\mathcal {B}}\circ T}
. This turns the space of bases into a left
GL
(
V
)
{\displaystyle {\text{GL}}(V)}
-torsor and a right
GL
(
n
,
K
)
{\displaystyle {\text{GL}}(n,K)}
-torsor. From a physical perspective, active transformations can be characterized as transformations of physical space, while passive transformations are characterized as redundancies in the description of physical space. This plays an important role in mathematical gauge theory, where gauge transformations are described mathematically by transition maps which act from the right on fibers.
== See also == Change of basis Covariance and contravariance of vectors Rotation of axes Translation of axes
== References ==
Dirk Struik (1953) Lectures on Analytic and Projective Geometry, page 84, Addison-Wesley.
== External links == UI ambiguity