analytical geometry

Question

We have an affine coordinate system and $3$ points given: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,1)$. I have to find a linear transformation, which depicts the points $A$, $B$, $C$ and $D$ accordingly into $B$, $C$, $A$ and $D$.

Answer 1

Besides observing that you need a permutation matrix, another option is to write your matrix $\mathbf{A} = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$ explicitly, and use the given data.

Using the fact that you should map your point $A$ to point $B$, you get

$\mathbf{A}\left[\begin{array}{c} 1 \\ 0 \\ 0\end{array}\right] = \left[\begin{array}{c} 0 \\ 1 \\ 0\end{array}\right]$

or equivalently,

$a_{11} = 0$, $a_{21} = 1$, $a_{31} = 0$.

Now use the rest of the data to find the other entries.