Let $G$ be a finite group. There exists a decomposition, as a $\mathbb{C}[G]$-module, $\mathbb{C}[G]=n_{1}V_{1}\oplus \cdots\oplus n_{r}V_{r}$, where the $V_{i}$ form a complete set of irreducible submodules of $\mathbb{C}[G]$, these submodules corresponding to the irreducible representations of $G$.
Why is the dimension of the representation $V_{i}$ equal to $n_{i}$, the number of copies of $V_{i}$ in the decomposition? I can prove that the space of $\mathbb{C}[G]$-module homomorphisms from $V_{1}$ to $V$ has dimension $n_{1}$, but I don't quite see how this implies that the dimension of the representation is also $n_{1}$.