# Prove there's a unique projective transformation that maps four points to four points

As the title states, the question is to prove that there's a unique Projective Transformation that maps four points of $$\mathbb{R^2}$$ to the projective plane. I tried defining the projective transformation as $$\bf{x'} = M\cdot \bf x$$ where $$M$$ is a $$3 \times3$$ matrix and defining four points $$x_i = \pmatrix{a_i\cr b_i}$$ but I don't think i'm getting anywhere. Any tips?

• Count degrees of freedom. – amd Sep 10 '19 at 8:24
• Alternatively, construct the transformation explicitly. One such construction is given here. You’ll find that you’ll need to impose some conditions on the points for your proposition to hold. – amd Sep 10 '19 at 10:50