GATE CS 2021 Set 2 Q37 — Sets, Relations and Functions

The given operation $s(P, Q) = \sum_{i=1}^n (P[i] \cdot Q[i])$ represents the standard inner product (dot product) of two vectors $P$ and $Q$ in $\mathbb{R}^n$. Here, $n = 10$.

We are given that:

• $\mathcal{L}$ is a set of $10$-dimensional non-zero real vectors.
• For every pair of distinct vectors $P, Q \in \mathcal{L}$, $s(P, Q) = 0$. This means the vectors in $\mathcal{L}$ are pairwise orthogonal.

A fundamental theorem in linear algebra states that any set of pairwise orthogonal non-zero vectors is linearly independent.

Since the vectors belong to a $10$-dimensional vector space $\mathbb{R}^{10}$, the maximum number of linearly independent vectors possible in this space is equal to its dimension, which is $10$.

Thus, the maximum cardinality of the set $\mathcal{L}$ is $10$.

Therefore, the correct option is B.