Let $u$ be a nonzero column vector understood as a matrix.
1) then there is an invertible matrix $A$ such that $Au=e_{1}$ where $e_1$ is the vector with one at the ith position and zeros at the others. Why? Is $A$ the product of elementary matrices?
2) Now, consider the part (1) and $uv^{t}$ is rank-one.Then $uv^t$ is similar to $e_{1}w^{t}$ where $w=A^{t}v$. Why?
I think you have a typo; we don't necessarily have $w = A^Tv$.
Note that $$ A(uv^T)A^{-1} = [Au][v^TA^{-1}] = [Au][(A^{-1})^Tv]^T = e_1[(A^{-1})^Tv]^T $$ So, if we take $w = (A^{-1})^Tv$, then the question makes sense.