Recover a unit vector from a unit normal and a linear transformation matrix

Question

The unit vector in the direction of $A\vec{u}$ where $A$ is an invertible ${m}\times{m}$ matrix and $n,u$ are $m\times1$ vectors is $$\vec{n} = \frac{A\vec{u}}{\|A\vec{u}\|}$$ how can I recover the unit vector in the direction of $\vec{u}$ if what I actually have is $A$ and $\vec{n}$?

Answer 1

Since $A$ is invertible, we have $$ \hat u = \frac{A^{-1}\vec n}{\|A^{-1}\vec n\|}. $$