The orbit of a non zero vector $v$ in $\mathbb{R}^n$ is $\mathbb{R}^n \setminus \{0\}$ Prove that given any non zero vectors $v$ and $w$ in $\mathbb{R}^n$, there exists an invertible $n\times n$ matrix $A$ such that $Av=w$.
(I don't know where to start. Any hints will be appreciated)

If $w$ is a scalar multiple of $v$, you could just let $A$ be a scalar matrix. That is, if $w=\lambda v$, then take $A=\lambda I$.
Otherwise $v$ and $w$ are linearly independent, and take $\{v,w,b_3,\ldots,b_n\}$ to be some basis of $\mathbb{R}^n$. Let $P$ be the $n\times n$ matrix $[v\ w\ b_3\ \ldots\ b_n]$. Then take $$A=P\left[\begin{array}{cc|c}0&1&\\1&0&\\\hline&&I\end{array}\right]P^{-1}$$
