Let $F$ be a field, and let $W=\begin{bmatrix} w_{1} &\cdots &w_{n}\cr 0 & 0 & 0\cr \vdots & \vdots & \vdots \cr 0 & 0 & 0\cr \end{bmatrix}$ be one-rank in $M_n(F)$. If $w_{1}=0$, then $W$ is similar to $E_{12}$ where $E_{12}$ denotes the matrix with 1 in the row $1$th and the column $2$th, and zero elsewhere.
If $w_{1}\neq 0$, then $W$ is similar to $w_{1}E_{11}$ where $E_{11}$ denotes the matrix with 1 in the row $1$th and the column $1$th, and zero elsewhere.
I want to find a basis for this Similarity, by changing of basis of matrices or Jordan basis. How?