Some cool stuff about linear algebra:

http://jakobschwichtenberg.com/adjoint-representation/

e.g.

http://jakobschwichtenberg.com/adjoint-representation/

e.g.

Aha! So that's what is special about an adjoint representation!It’s often a good idea to look at the Lie algebra of a group to study its properties, because working with a vector space, like $$ TeG $$, is in general easier than working with some curved space, like $$ G $$. An important theorem alled Ado’s Theorem, tells us that every Lie algebra is isomorphic to a matrix Lie algebra.This tells us that the knowledge of ordinary linear algebra is enough to study Lie algebrasbecause every Lie algebra can be viewed as a set of matrices.

A natural idea is now to have a look at the representation of the group $$ G $$ on the only distinguished vector space that comes automatically with each Lie group: The representation on its own tangent vector space at the identity $$ TeG $$, i.e. the Lie algebra of the group!

In other words, in principle, we can look at representations of a given group on any vector space. But there is exactly one distinguished vector space that comes automatically with each group: Its own Lie algebra.This representation is the adjoint representation.

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