are linear combinations of the columns of M; so for some 2×2 nonsingular matrix Λ, In particular, rows i and j of M′ and M are related by Therefore, the determinant of the left side 2×2 matrix equals the product of the determinants of the right side 2×2 matrices, the latter of which is a fixed scalar, det Λ. Furthermore, all six 2×2 subdeterminants in M cannot be zero because the rank of M is 2. Plücker map Denote the set of all lines (linear images of P1) in P3 by G1,3. We thus have a map: where Dual coordinates Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a0:a1:a2:a3) and (b0:b1:b2:b3), respectively. (The first plane equation is Σk akxk=0, for example.) The dual Plücker coordinate pij is {| |- | || |} Dual coordinates are convenient in some computations, and they are equivalent to primary coordinates: Here, equality between the two vectors in homogeneous coordinates means that the numbers on the right side are equal to the numbers on the left side up to some common scaling factor . Specifically, let (i,j,k,ℓ) be an even permutation of (0,1,2,3); then Geometry To relate back to the geometric intuition, take x0 = 0 as the plane at infinity; thus the coordinates of points not at infinity can be normalized so that x0 = 1. Then M becomes and setting and , we have and . Dually, we have and . Bijection between lines and Klein quadric Plane equations If the point z = (z0:z1:z2:z3) lies on L, then the columns of are linearly dependent, so that the rank of this larger matrix is still 2. This implies that all 3×3 submatrices have determinant zero, generating four (4 choose 3) plane equations, such as The four possible planes obtained are as follows. Using dual coordinates, and letting (a0:a1:a2:a3) be the line coefficients, each of these is simply ai = pij, or Each Plücker coordinate appears in two of the four equations, each time multiplying a different variable; and as at least one of the coordinates is nonzero, we are guaranteed non-vacuous equations for two distinct planes intersecting in L. Thus the Plücker coordinates of a line determine that line uniquely, and the map α is an injection. Quadratic relation The image of α is