-vector space of dimension , and be the canonical projection that maps a nonzero vector to the vector line that contains it. For every frame of , there exists a basis of such that the frame is and this basis is unique up to the multiplication of all its elements by the same nonzero element of . Conversely, if is a basis of , then is a frame of It follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). It is sometimes called the first fundamental theorem of projective geometry. Every frame allows to define projective coordinates, also known as homogeneous coordinates: every point may be written as ; the projective coordinates of on this frame are the coordinates of on the base It is not difficult to verify that changing the and , without changing the frame nor p(v), results in multiplying the projective coordinates by the same nonzero element of K. The projective space has a canonical frame consisting of the image by of the canonical basis of (consisting of the elements having only one nonzero entry, which is equal to 1), and . On this basis, the homogeneous coordinates of are simply the entries (coefficients) of the tuple . Given another projective space of the same dimension, and a frame of it, there is one and only one homography mapping onto the canonical frame of . The projective coordinates of a point on the frame are the homogeneous coordinates of on the canonical frame of . Central collineations In above sections, homographies have been defined through linear algebra. In synthetic geometry, they are traditionally defined as the composition of one or several special homographies called central collineations. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent. In a projective space, P, of dimension , a collineation of P is a bijection from P onto P that maps lines onto lines. A central collineation (traditionally these were called perspectivities, but this term may be confusing, having another meaning; see Perspectivity) is a bijection α from P to P, such that there exists a hyperplane H (called the axis of α),