Example of application of creation/annihilation operators in matrix form I was wondering how it would sound like the creation/annihilation of particles that we usually do in the context of Dirac formalism, with matrices and vectors. As a reminder we know that:
\begin{equation}
a^{-}\left|n\right\rangle =\sqrt{n}\left|n-1\right\rangle 
\end{equation}
and
\begin{equation}
a^{+}\left|n\right\rangle =\sqrt{n+1}\left|n+1\right\rangle 
\end{equation}
The operators are defined as:
\begin{equation*}a^{-}=\left(\begin{array}{ccccc}
0 & 1 & 0 & 0 & ...\\
0 & 0 & 1 & 0 & ...\\
0 & 0 & 0 & 1 & ...\\
0 & 0 & 0 & 0 & ...\\
... & ... & ... & ... & ...
\end{array}\right)
\end{equation*}
and
\begin{equation*}a^{+}=\left(\begin{array}{ccccc}
0 & 0 & 0 & 0 & ...\\
1 & 0 & 0 & 0 & ...\\
0 & 1 & 0 & 0 & ...\\
0 & 0 & 1 & 0 & ...\\
... & ... & ... & ... & ...
\end{array}\right)
\end{equation*}
So how is the form of the ket $\left|n\right\rangle$ in terms of a vector? It must a tensor product of vectors for sure. Moreover the matrix $a^{-}$ (or $a^{+}$) times $\left|0\right\rangle$ (probably a vector with all its entries equal to $0$?) must be something that gives a vector with an entry increased of $1$. But, to be rigorous, a prefix should be added to the matrix in order to indicate that it is acting on a precise factor of the tensor product which made up $\left|n\right\rangle$.
An example of a multiplication between a creation/annihilation matrix for a vector would be highly appreciated.
If the question is not clear let me know!
 A: As mentioned in the comments. Your matrix representation of the creation and annihilation operators is incorrect. This is easy to see since
\begin{align} 
a ^\dagger _{ nm} & = \left\langle n \right|  a ^\dagger \left| m \right\rangle
\\ 
& = \sqrt{ m + 1 } \delta _{ n , m + 1 }.
\end{align} 
Thus we have,
\begin{equation} 
\left( \begin{array}{cccc} 
0 & 0 & 0 & ...\\  
1 & 0 & 0 &...\\  
0 & \sqrt{2}  & 0 &... \\  
0 &0   & \sqrt{3}  & \ddots\\  
\vdots &...  & \ddots & \ddots   
\end{array} \right) 
\end{equation} 
The kets in this space are simple:
\begin{equation} 
\left| n \right\rangle = \left( \begin{array}{c} 
0 \\  
\vdots \\  
0 \\ 
1 \\  
0 \\ 
\vdots \\ 
0
\end{array} \right) 
\end{equation} 
where $1$ is at the $ n +1 $'th spot down the vector. Its easy to see that acting on this vector we get a result consistent with the relation for the raising operator above. 
The vacuum state is then just given by 
\begin{equation} 
\left| 0 \right\rangle = \left( \begin{array}{c} 
1 \\  
0 \\  
\vdots  \\  
0 
\end{array} \right) 
\end{equation} 
