Let x1,..., xk E R2 and assume these points do not all lie on a line. Our goal i
ID: 2250528 • Letter: L
Question
Let x1,..., xk E R2 and assume these points do not all lie on a line. Our goal in this problem is to prove that there exist yi, ype R2 so that (xi, ,%) = (Y1,Y2) (So, just like vector spaces, you only need to use 2 vectors to generate a lattice in 2 dimensions.) Fill in the logic in the following outline of a proof. (a) Over all points in the set (Xi, ,Xk) assume that y1 is closest to 0 without being 0. Let L be the line containing the points 0 and y1. Now, over all points in A(x1,... , xk) which are not on the line L, assume that y2 is closest to L. Consider the parallelogram P with vertices 0, yi, V2, yi + y2. Show that the only points in A(x1,... , xx) which lie on or inside this parallelogram are the vertices (b) The lattice A(y1, y2) divides the plane into parallelograms which are copies of P. Show that these other parallelograms also have the property that only their vertices are in the set (x1, . . . , xk).Explanation / Answer
Here given sequence of vectors in the set are /(x1,x2......xk) £R2 and these points do not lie on the same line.So,we can form any two dimensional figures with these points like square,parallelogram etc.
Here,Other side we have only two vectors y1 and y2 to generate a lattice in 2 dimensions.
(a). Given that,y1 is closest to the point 0 and both are lie on the same line L to form parallelogram and some of the points in the sequence x1....xk lie on this line and in side this parallelogram.Total 4 vertices of parallelogram are 0,y1,y2,y1+y2.The points which lie on (or) inside the two dimensional figures are only the vertices of those figures.
So,Here it is proved that the points in the sequence x1.....xk which are lie on (or) inside the parallelogram formed with two vectors y1,y2 are the only vertices of that parallelogram. So,from this It is proved that
/(x1,..xk)=/(y1,y2)
0/----------------L---------------/y1
y2/------------------------------------/ (y1+y2)
(2). we can diivde the above lattice /(y1,y2) in to many nnumber of parallelograms From those points in different directions and which are similar to the above formed parallelogram p that means copies of p.and those formed parallelograms also contain the points in the set x1...xk.So,These points which are lie on those parallelograms also vertices of those parallelograms because some points definitely lie on (or) inside those parallelograms.
So,Finally from above explanation also,it is proved that
/(x1...xk)=/(y1,y2).
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.