GATE CS 2021 Set 2 — Question 47
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Engineering Mathematics → Linear Algebra → Vector Spaces & Linear Maps
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Question
For two -dimensional real vectors and , the operation is defined as follows:Let be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors , . What is the maximum cardinality possible for the set ?
Correct answer
(B) 10
Solution
The operation is the standard dot product (inner product) in . The condition for distinct implies that the vectors in the set are pairwise orthogonal. It is a well-known property in linear algebra that any set of pairwise orthogonal non-zero vectors is linearly independent. Since the vectors are 10-dimensional, they are elements of the vector space . The dimension of is 10, which means the maximum number of linearly independent vectors in this space is 10. Therefore, the maximum cardinality of the set is 10.
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