Find the projective transformation $\tau \left ( \left [ 0,0,1 \right ] \right )=\left [ 0,1,0 \right ], \tau ([0,1,0])=[0,1,1],\tau ([1,0,0])=[1,1,1], \tau ([1,1,2])=[1,1,0]$. And show that such a projective transformation is unique.
What is a good way to solve this problem? Thanks a lot.
This answer of mine shows the general approach of how to compute a projective transformation in the plane given four points and their images. Just plug your numbers into that algorithm and you end up with the required transformation matrix.
How to show that a projective transformation is unique depends on what definitions you have established. Usually you assume the contrary: suppose you had two distinct transformations with the desired properties. Then you could perform one and the inverse of the other and would end up with a combined transformation which fixes the four preimage points but which is not the identity.
Consider one dimension lower first: given a projective transformation of the projective line, which fixes three points. Show that it is the identity. You can do that by observing that the cross ratio of any four points remains fixed under projective transformations, and that knowing the cross ratio with respect to three given points already uniquely defines any point on the line.
To take this approach to the plane, you can use your four points to establish something like a projective coordinate system. Usually you would use $[0,0,1]$ as the origin, $[1,0,0]$ as the point at infinity in the $x$ direction, $[0,1,0]$ at infinity in the $y$ direction, $[1,0,1]$ as the unit step in the $x$ direction and $[0,1,1]$ as the unit step in the $y$ direction. The last two can be seen as projections of a single point $[1,1,1]$, i.e. join $[1,1,1]$ with $[0,1,0]$ and intersect the resulting line with the $x$ axis to obtain the unit step there, and likewise for $y$. So these four points $[1,0,0],[0,1,0],[0,0,1],[1,1,1]$ define this coordinate system. Any other point can be projected onto the axes and cross ratios computed there will give its $x$ and $y$ coordinate.
The choice of coordinates in the basis I just used was just for convenience. Any other four points can serve the same purpose, as long as no three of them are collinear. Computing the coordinates of a point with respect to such a projective basis would mean performing some joins and intersections and computing some cross ratios. These coordinates cannot change under projective transformations, and uniquely define each point in the plane. Both of these properties can be shown fairly easily.
Once you have that, showing that a projective transformation is unique boils down to showing that no three of its preimage points are collinear. In fact, if they were collinear, the matrix you'd get from the algorithm I suggested would be degenerate, i.e. not have full rank.
