A difficulty in understanding the definition of "Spaces of Matrix Elements." The definition is given below:
If the definition of the Spaces of Matrix Elements is as given below:


 
But I do not understand why:
1- Any linear combination of matrix elements can be expressed invariantly (without using coordinates) as given in Eq.(3), could anyone explain this for me please?
2-Also I do not understand the paragraph under eq.(3), and why it is c_{ji} and not c_{ij}, could anyone explain this for me with a concrete example please?
3-Could anyone give me a concrete example describing the difference between the space of matrix elements of the representation $T$ and the matrix elements of $T$?
Thank you!
 A: 
1- Any linear combination of matrix elements can be expressed invariantly (without using coordinates) as given in Eq.(3), could anyone explain this for me please?

The point is that the operation "trace" is independent on the coordinates. That is if $A$ is a matrix and $B$ is obtained from $A$ by changing basis then $B=PAP^{-1}$ and $\text{trace}(A)=\text{trace}(B)$. 

2-Also I do not understand the paragraph under eq.(3), and why it is $c_{ji}$ and not $c_{ij}$, could anyone explain this for me with a concrete example please?

It's minor and technical, if you don't take the transposed one then you would get $\sum_{i,j} c_{j,i}T_{i,j}$ instead. Let me explain (I'm going to use an unusual letters for the indices in the beginning so you will see the correct picture in the end): If you take $C=[c_{j,l}]$ and $T(g)=[T_{j,l}(g)]$ then $$CT(g)_{j,l} = \sum_{i=1}^n c_{j,i}T_{i,l}(g)$$ and so the trace would be
$$\sum_{j=1}^n CT(g)_{j,j} = \sum_{j=1}^n \sum_{i=1}^n c_{j,i}T_{i,j}(g)$$
But as you can see the indices are not the way the author wanted them to be, they're $c_{j,i}T_{j,i}$ instead of $c_{j,i}T_{j,i}$. In order to overcome this minor issue the author decided to multiply by the transposed matrix. 

3-Could anyone give me a concrete example describing the difference between the space of matrix elements of the representation T and the matrix elements of T?

Before I answer this question let me explain:
Given any $T:G\rightarrow GL(V)$ the matrix coefficients are functions from $G$ to $\mathbb{C}$. The space of matrix elements are not just matrix coefficients, but also their linear combinations.
Consider the trivial example: Take $G$ to be any group, and $T:G\rightarrow GL(V)$ be trivial. In that case no matter which base you choose, there is only one matrix coordinate which is $\varphi(g)=n$ (A constant map from $G$ to $\mathbb{C}$, where $n$ is the dimension of $V$ which is also the trace of the identity matrix). However the space of matrix elements contains any linear combination as well, in particular it contains $2\varphi(g)=2n$ (which is another constant map with different constant).
