For two n-dimensional real vectors P and Q, the operation a(P, Q) is defined as follows: a(P, Q) = sum_{i=1}^{n} (P_i * Q_i) Let C be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P, Q in C, a(P, Q) = 0. What is the maximum cardinality possible for the set C? A. 9 B. 10 C. 11 D. 100

GATE 2021 · Discrete Mathematics · Set Theory · medium

Answer: B. 10

1. Identify a(P,Q) as the dot product: This is the standard Euclidean inner product (dot product) in R^10. The condition a(P,Q) = 0 means P and Q are orthogonal.
2. Mutually orthogonal non-zero vectors are linearly independent: Suppose c_1*v_1 + ... + c_k*v_k = 0. Taking the dot product with v_j: c_j * a(v_j, v_j) = 0. Since v_j != 0, a(v_j, v_j) = sum v_{ji}^2 > 0, so c_j = 0. This holds for all j, proving independence.
3. Dimension bound and achievability: Since any mutually orthogonal non-zero set is linearly independent in R^10, and R^10 has dimension 10, |C| <= 10. The standard basis {e_1, e_2, ..., e_10} is a set of 10 mutually orthogonal non-zero vectors, achieving the maximum. So max |C| = 10.