I am in difficulty with the following problem:
Let $u,v\in\mathbb{R}^n$ such that $u\ne v$. Constructing matrices $A,B\in\mathbb{R}^{n\times n}$ and vectors $a,b\in\mathbb{R}^n$ such that $$ Ae_1=v-u, \quad Aa=u, $$ $$ B(u+b)=0, \quad B(v+b)=e_1. $$ where $e_1=(1,0,\ldots,0)\in\mathbb{R}^n$.
In my opinion this problem has a relation of rotation, transformation and dilation of coordinate system which transform $u$ into $0$ and $v$ into $e_1$ and reverse.
Thank you for all kind help.
$A$ can simply be chosen as $A=\begin{pmatrix}v-u&u&0\end{pmatrix}$ if we let $a = e_2$
To achieve $B(u+b) = 0$ we simply pick $b = -u$ so last but not least we need to find $b$ such that $B(v-u) = e_1$, which is achieved for $ B = \begin{pmatrix} \frac{1}{v^Tv - u^Tu}(v^T + u^T) \\0 \end{pmatrix}$
Note that $B$ is well-defined since we assume $v\neq u$.