Dual Representation [The Physics Travel Guide]

--- advanced_tools:group_theory:representation_theory:dual_representation [2022/07/03 20:38]
edi [Concrete]
+++ advanced_tools:group_theory:representation_theory:dual_representation [2023/02/20 18:54]
edi [Intuitive]
@@ Line 2: / Line 2: @@
 <tabbox Intuitive>
-For any representation, we can construct a dual representation. To do that we let the representation act on the dual vector space instead of the original vector space. The dual vector space consists of covectors, that is, linear functions that map vectors to scalars.
+For any representation, there is a dual representation, which acts on the dual vector space instead of the original vector space. The dual vector space consists of covectors, which are linear functions from the vectors to scalars.
-In some cases, the dual representation is equivalent to the original one (e.g. for representations by orthogonal matrices).
+In some cases, the dual representation and the original representation are the same (e.g. for representations with orthogonal matrices).
-For unitary representations, the dual representation is also the complex-conjugate representation.
+For unitary representations, the dual representation is the same as the complex-conjugate representation.
 <tabbox Concrete>