Burnside's theorem proof help

I'm currently reading the book "Representation Theory Vol. 1" by Kirillov, Bernstein and Arnold. In page 9 of the book the Burnside theorem is partially proven, I'll state it here so there is no confusion:

Let $G$ be a finite group, $|G| = N$; let $T_1,..., T_a$ be the full set of irreducible representations of $G$, let $dim V_i = n_i$. Then $N = \sum_i n_i^2$

The given proof starts with the regular representations. Let $L(G) = Maps(G,K)$ the space of functions on $G$. For the basis of $L(G)$ we take $\delta - functions$ $\delta_g (h) = 1$ if $g = h$ and 0 otherwise. The hermitian product is also introduced:

$$(f_1, f_2) = \sum_g f_1(g) \overline{f_2(g)}$$

The book asks to prove that the delta functions are an orthonormal basis which I have done already. The the idea is to construct another basis such that to each $n$ dimensional irreducible representation, there corresponds $n^2$ elements of this new basis. Let $T$ be an irreducible representation, $T(g)$ can be considered as a matrix valued function on $G$, then each element of the matrix $T(g)_{ij}$ is a numerical function. The book asks to prove that this functions are linearly independent. But I'm stuck with this. Let us have $\alpha_{ij}$ such that

$$\sum_{ij} \alpha_{ij} T_{ij}(g) = 0$$

for all $g$. I can't prove that $\alpha_{ij} = 0$. I'm guessing I have to use the property of irreducibility of $T$ but I haven't being able so any help with whis would be appreciated.

After this it is mentioned that to complete the proof, it needs to be proven that $L(G)$ is the direct sum of $span(T_{ij})$ for various T. And that $Span(T_{ij})$ is orthogonal to $Span(U_{kl})$ for $T$ and $U$ not equivalent representations. Here I think I need to use schur's lemma but I'm having difficulty seeing how. So any help on the direction to prove this would also be appreciated!