Throughout this post is a finite group.

**First Construction**. Suppose that are two representations of It is straightforward to show that the map defined by

for all and is a representation of It is also clear that

**Second Construction**. Suppose that are two representations of It is easy to see that the map defined by

for all and and then of course extended linearly to all elements of is a representation of In this case we have

**Third Construction**. Suppose that is a representation of Let be the dual space of i.e. the set of all linear maps The map defined by

for all and is a representation of We will call the **dual representation** of Clearly

In the next post I will discuss the dual representation in more details.

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