#### Gist

**Representation** is just a homomorphism mapping

(1)

where is an operator on linear vector space , such that

(2)

**Dimension of the representation**: is the dimension of the vector space .

**Faithful**: If the homomorphism is also an isomorphism.

**Degenerate representation**: The one which is not Faithful.

**Matrix Representation**: is the case that is a matrix and the matrix multiplication preserve the group multiplication.

#### Analysis

The most important concept is mapping. In fact, we device group representation for the purpose of elaborating the operations(group) in a mathematical way, thus we can write it down and discussion it in math symbols.

Problems

3.1 Consider the six dihedral group . Let be the 2-dimensional Euclidean space spanned by and . Write down the matrix representation of elements of on with respect to this Cartesian basis.

Just right down the matrix which can transform the and into and as what the transformation in . That is the matrix representation is the matrix in

(3)

Based on this Figure, we can have

It seems that the solution to Problems 3.1 is wrong.

The Eq.(3) should be

(1)

where, we have written as , as . And the representation matrices should be

However, the solution by Michael Aivazis has the same form in my previous solution.