Implicit Scheme
We obtain now
\begin{equation*}
u_{i,j-1}= -\alpha u_{i-1,j}+(1-2\alpha)u_{i,j}-\alpha u_{i+1,j}.
\end{equation*}
Here
u_{i,j-1} is the only unknown quantity.
Defining the matrix
\mathbf{A}
\begin{equation*}
\mathbf{A}=\begin{bmatrix}1+2\alpha&-\alpha&0& 0 &\dots\\
-\alpha&1+2\alpha&-\alpha & 0 & \dots \\
\dots & \dots & \dots & \dots &\dots \\
\dots & \dots & \dots & \dots & -\alpha \\
0 & 0 &\dots &-\alpha& 1+2\alpha\end{bmatrix},
\end{equation*}
we can reformulate again the problem as a matrix-vector multiplication
\begin{equation*}
\mathbf{A}V_{j} = V_{j-1}
\end{equation*}