Let n ≥ 4 be an integer. Let S be a n × n square in the plane whose vertices are the lattice points (a, b),(a + n, b),(a, b + n),(a + n, b + n) for some integers a and b. The interior of S is defined to be all points in S which are not vertices of S and not contained in any of the 4 sides of S. Find the smallest value of N that guarantees the following: For any set A of N lattice points contained in the interior of S, there are two elements (x, y),(x'.y') of A such that x-x'=y'-y  Justify your answer.

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.