Linear Transformations With Quaternions

The set of all quaternions  
\[q=a+b \mathbf{i}+c \mathbf{j}+d \mathbf{k}\]
  where  
\[a, \; b, \; c, d \in \mathbb{R}\]
  with
\[\mathbf{i}^2= \mathbf{j}^2 = \mathbf{k}^2 =-1\]

\[ij=k, \; jk=i, \; ki=j, \; ji=-k, \;kj=-i, \; ik=-j\]

forms a vector space over  
\[\mathbb{R}\]
  with basis  
\[1, \; \mathbf{i}, \; \mathbf{j}, \; \mathbf{k}\]
.
We can represent a vector in
\[\mathbf{v}= \begin{pmatrix}x\\y\\z\end{pmatrix} \in \mathbb{R}^3\]
  as the quaternion  
\[q=x \mathbf{i}+ y \mathbf{j}+z \mathbf{k}\]
. Let  
\[\mathbf{v} \in \mathbb{R}^3\]
  and let  
\[q=a+b \mathbf{v}\]
  be a unit quaternion, so that  
\[\sqrt{a^2+b^2}=1\]
.
Let  
\[\mathbf{w} \in \mathbb{R}^3\]
  and consider the mapping  
\[T: \mathbf{w}q \mathbf{x} \bar{q}=(a+b \mathbf{v})\mathbf{w}(a-b \mathbf{v})\]
.
This is also a vector in  
\[\mathbb{R}^3\]
  so the mapping above is a linear transformation from  
\[\mathbb{R}^3\]
  to  
\[\mathbb{R}^3\]
.

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