Why E's representation marix is unit matrix

AI Thread Summary
In group representation theory, the identity element's representation matrix is the unit matrix because it must satisfy the property that multiplying any matrix A by the identity matrix E results in A. This means that for the identity element e, represented by matrix E, the equations AE = A and EA = A hold true for all group members represented by matrix A. The identity matrix is the only matrix that fulfills these conditions, confirming that E must be the unit matrix. The discussion seeks a formal proof of this concept, emphasizing the foundational role of the identity element in group representations. Understanding this relationship is crucial for deeper insights into the structure of group representations.
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In group representation theory, identity element's representation matrix is unit matrix. But why? Could you give me an exact explanation? Thany you.
 
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Group representation theory essentially boils down to using matrices to represent group elements. In other words, a group is described as a set of linear transformations on vector spaces. In this respect it is obvious that the identity element will be the identity matrix.
 
I know this, but I don't know how to prove that exactly
 
The group identity has the property that ae= a and ea= a for any group member a. Suppose e is "represented" by the matrix E and a by the matrix A. Then you must have AE= A and EA= A, for any matrix, A, in the group representation. Therefore E= ?
 
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