Recall that Z_2 = {0,1}. with addition and multiplication defined modulo 2. In t
ID: 1720164 • Letter: R
Question
Recall that Z_2 = {0,1}. with addition and multiplication defined modulo 2. In the model Z2P^2, points are lines through (0,0,0) in (Z_2)^3, and lines are planes through (0,0,0) in (Z_2)^3. A line through (0.0.0) in (Z_2)^3 can be written parametrically in the form t(p,q, r), t epsilon Z_2. where (p,q, r) is a nonzero vector in (Z_2)^3. Similarly, a plane through (0,0,0) in (Z_2)^3 can be written parametrically in the form s(a,b,c) + t(d,e,f), s,t epsilon Z_2, where (a,b,c) and (d, e, f) are linearly independent vectors in (Z_2)^3. in both cases, the parameters can only take values in Z_2. The following are equations of planes through (0,0,0) in (Z_2)^3. Write each plane in parametric form. (i) y = 0 (ii) x + z = 0 Consider an arbitrary plane through (0,0,0) in (Z_2)^3. How many distinct points in (Z_2)^3 does it contain? How many distinct lines through (0,0,0) does it contain? Your answer should include an explanation of why the points or lines are distinct, and why they exhaust all possibilities. Conclude that every line in Z_2P^2 contains exactly three distinct points in Z_2P^2.Explanation / Answer
(a) (i) The plane y =0,. We take two independent solutions
(1,0,0) and (0,0,1)
SO the parametic form of this plane is given by
s(1,0,0) + t (0,0,1).
(ii) For the plane X+Z =0, we may take the independent vectors
(1,0,1) and (0,1,0).
So the parametric form of this plane is given by
s(1,0,1) + t (0,1,0).
(b) Any plane in Z23 is just a two dimensional subspace . So it contains 22 =4 points.
Any line corresponds to a non-zero vector in Z23 .As there are 7 such vectors , there are 7 lines .
(c) As a line in Z2 P2 is a plane in Z23 it follow that a line can be considered as a plane in Z23 with the origin omitted.
Thus every line contains exactly 3 points
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