{(х, у, г) € R$I(x, у, 2) # (0, 0, 0)}. Define ~ on T by (x], У1, Z1) Ax2, y1 = Ay2, 12. Let T (x2, y2, z2) if there exists a nonzero real number A such that x1 = and z1 = Az2. (a) Show that ~ is an...

Please answer part h.

Extracted text: {(х, у, г) € R$I(x, у, 2) # (0, 0, 0)}. Define ~ on T by (x], У1, Z1) Ax2, y1 = Ay2, 12. Let T (x2, y2, z2) if there exists a nonzero real number A such that x1 = and z1 = Az2. (a) Show that ~ is an equivalence relation on T. (b) Give a geometric description of the equivalence class of (x, y, z). The set T/ (x, y, z) is denoted by [x, y, z], and is called a point. ~ is called the real projective plane, and is denoted by P2. The class of (c) Let (a, b, c) € T, and suppose that (x1, y1, Z1) axi + byi + czi = 0, then ax2 + by2 + cz2 = 0. Conclude that (x2, y2, z2). Show that if L%3D ([x, у, z] € P|аx + by + cz %3D 0} is a well-defined subset of P2. Such sets L are called lines. (d) Show that the triples (a1, b1, c1) e T and (a2, b2, c2) E T determine the same line if and only if (a1, b1, c1) ~ (a2, b2, c2). (e) Given two distinct points of P?, show that there exists exactly one line that contains both points. (f) Given two distinct lines, show that there exists exactly one point that belongs to both lines. (g) Show that the function f : R2 → P² defined by f (x, y) = [x, y, 1] is a one-to- one function. This is one possible embedding of the “affine plane" into the projective plane. We sometimes say that P2 is the "completion" of R?. (h) Show that the embedding of part (g) takes lines to "lines." (i) If two lines intersect in R2, show that the image of their intersection is the inter- section of their images (under the embedding defined in part (g)). (j) If two lines are parallel in R², what happens to their images under the embedding into P2?