Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:
T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1
When you are transforming basis $\displaystyle e_i $ vectors the matrix will be
$\displaystyle T=\begin{bmatrix}T(e_1) && T(e_2) && T(e_3) && T(e_4) \end{bmatrix}$
Or in your case
$\displaystyle T=
\begin{bmatrix}T(v_1) && T(v_2) && T(v_3) && T(v_4) \end{bmatrix}=
\begin{bmatrix}v_2 && v_3 && v_4 && v_1 \end{bmatrix}$
The j'th column of the matrix will consist of the coefficients of T(v_j) when expressed as a linear combination of the four basis vectors.
For example, $\displaystyle Tv_1=v_2=0v_1+1v_2+0v_3+0v_4$, so the first column of the matrix will consist of 0,1,0,0.
Thus the matrix will be $\displaystyle \begin{bmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\ end{bmatrix}.$