7. How much freedom in making projective transformations of the plane?

A homogeneous linear transformation is a special instance of an affine transformation, and an affine transformation is a special instance of a projective transformation.

With a homogeneous linear transformation we can map two given vectors, not along the same line, to two other given vectors. In terms of points, the origin must go to the origin, but we could take two other points, not on the same line through the origin, to two given points.

With an affine transformation we can take a triangle to a triangle. (The three points of the first triangle must not be collinear.)

With a projective transformation of P$_2 \rightarrow$   P$_2$, we can take a quartet to a quartet.

Fact. The projective transformation taking one given quartet to another is unique.

Note. Its matrix is not unique, since multiplying the whole matrix by a nonzero scalar has no effect on the transformation. That's the only way the matrix can vary, though. In particular, you cannot multiply each row of the matrix by a separate nonzero scalar. That would not change the images of $X$, $Y$, and $O$, but it does change the images of other points, such as $E$.