Coordinate change in $n$-dimensional space

Question

I have a unit vector in an $n$-dimensional space, $V = [v_1, v_2, ..., v_n]$. I am looking for a transformation $R$, $\;W = R * V$, which could give me back $\;W = [1, 0, ..., 0]$. Any idea?

Answer 1

If I understand your question correctly, perhaps $$ R= \begin{bmatrix}1/v_1 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0\end{bmatrix}?$$

Answer 2

Let $e_1 $ be the unit vector with one in the first position.

Note that $v^T v = \|v\|^2$.

Consider $R= {1 \over \|v\|^2} e_1 v^T$. Note that $R v = e_1$.

This assumes $v\neq 0$, of course.

Answer 3

I think that the reason this even came up as an issue is that the transformation might not be unique. Let's use the example of a 3D space, where you want to rotate $(0,1,0)$ into $(1,0,0)$. You can rotate around the $(0,0,1)$ axis by $-90^\circ$, or rotate around the $(0,0,-1)$ axis by $90^\circ$. But you can rotate around the $(1,1,0)$ axis by $\pm180^\circ$. In fact you can create a cone with $(h,h,l)$ axis ($h,l\in\mathbb R$), with $(0,1,0)$ on the surface, that you can rotate into $(1,0,0)$. That's because $(0,0,0)$ is the vertex of the cone, and the distance from the given axis to the $(0,1,0)$ and $(1,0,0)$ is the same.

Answer 4

In fact, I have an mxn matrix P where each row is a vector in the n-dimensional space. I have another row vector in the n-dimensional space V = [v1, v2, ..., vn]. I want to transform my original coordinate system through R such that transformed V (=W) becomes one of the basis of the new space, i.e. W = R * V; e.g. W = [1, 0, ..., 0]. At the end, I want to transfer my original matrix P using R.

It does not seem to be the right choice to use matrix R with R₁₁=1/v₁ and all other R_ij=0 for i=1,..,n; j=1,..,n. I'd appreciate it if anyone could help...