GATE CS 2021 Set 2 Q.47 (CS Q.37): Linear Algebra

For two $n$ -dimensional real vectors $P$ and $Q$ , the operation $s(P, Q)$ is defined as follows:

s(P, Q) = \sum_{i=1}^n (P_i \cdot Q_i)

Let $\mathcal{L}$ be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors $P, Q \in \mathcal{L}$ , $s(P, Q) = 0$ . What is the maximum cardinality possible for the set $\mathcal{L}$ ?

100

Official answer and solution

Official answer: B
From: Official GATE 2021 answer key

Solution

Recognize $s(P, Q)$ as the Euclidean dot product and use the theorem that mutually orthogonal non-zero vectors in an inner product space are linearly independent.

The operation $s(P, Q) = \sum_{i=1}^n P_i Q_i$ is the standard inner product (dot product) in $\mathbb{R}^n$ .

The set $\mathcal{L}$ consists of non-zero vectors in $\mathbb{R}^{10}$ such that $s(P, Q) = 0$ for every distinct pair $P, Q \in \mathcal{L}$ . This means all vectors in $\mathcal{L}$ are mutually orthogonal.

A well-known fundamental theorem of linear algebra states that any non-empty set of pairwise orthogonal non-zero vectors in an inner product space is linearly independent.

In the vector space $\mathbb{R}^{10}$ , the dimension is 10. The maximum number of linearly independent vectors in an $n$ -dimensional space cannot exceed the dimension of the space ( $n = 10$ ).

Therefore, the maximum cardinality possible for $\mathcal{L}$ is 10, which matches option B.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 47 of 65, printed as CS Q.37
Section: Computer Science and Information Technology

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