TITLE: Can we find the inverse for a vector?
QUESTION [10 upvotes]: Can we invert a vector like we do with matrices, and why?
I didn't see in any linear algebra course the concept of the "vector inverse", and I was wondering if there is any such thing, and if not, why.

REPLY [4 votes]: In some applications (e.g. extrapolation methods), one often considers what is called the Samelson inverse of a vector:
$$\mathbf v^{(-1)}=\frac{\bar{\mathbf v}}{\bar{\mathbf v}\cdot\mathbf v}=\frac{\bar{\mathbf v}}{\|\mathbf v\|^2}$$
(where the bar denotes complex conjugation), which can be easily shown to satisfy $\mathbf v^{(-1)}\cdot\mathbf v=\mathbf v\cdot\mathbf v^{(-1)}=1$ (the Moore-Penrose conditions in vector garb). (As usual, $\mathbf v\neq\mathbf 0$.)
Some references that use this inverse include this, this, and this.