Let $A=C([0,1])$ be an abstract $C^{*}$-algebra. Let $(t_i)_{i=1}^{\infty}$ be a dense sequence in $A$, using this, we define $\phi:A\to\mathcal{B}(\ell^2(\mathbb{N}))$ as below: $$\phi(a)=\begin{bmatrix} a(t_1) & 0 & 0 & \dots\\ 0 & a(t_2) & 0 &\dots\\ 0 & 0 & a(t_3) &\dots\\ \vdots & \vdots & \ddots\end{bmatrix}$$
Show that $\phi$ is an injective $*-$homomorphism.
As a newcomer to $C^{*}$-algebra, I don't quite get the idea of solving this problem. Any help please?