For two n-dimensional real vectors P and Q, s(P,Q) = sum of P[i]*Q[i]. Let L be a set of 10-dimensio

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Question

For two n-dimensional real vectors P and Q, s(P,Q) = sum of P[i]*Q[i]. Let L be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q in L, s(P,Q) = 0. What is the maximum cardinality possible for L?

Accepted Answer

Correct Answer: (D) 10Concept -- Orthogonal Vectors and Dimension (from Gilbert Strang, Linear Algebra and Its Applications):s(P,Q) = P.Q (dot product) = 0 for distinct P,Q in L means all vectors in L are mutually orthogonal (pairwise orthogonal, forming an orthogonal set).A key theorem from linear algebra: In R^n, the maximum size of a set of mutually orthogonal non-zero vectors is n. This is because mutually orthogonal non-zero vectors are linearly independent, and in R^n, any linearly independent set can have at most n vectors.For 10-dimensional vectors (n=10): maximum number of mutually orthogonal non-zero vectors = 10. An example is the standard basis {e1, e2, ..., e10} where each ei is the vector with 1 in position i and 0 elsewhere. These are pairwise orthogonal and all non-zero.Maximum cardinality of L = 10.

For two n-dimensional real vectors P and Q, s(P,Q) = sum of P[i]*Q[i]. Let L be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q in L, s(P,Q) = 0. What is the maximum cardinality possible for L?

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For two n-dimensional real vectors P and Q, s(P,Q) = sum of P[i]*Q[i]. Let L be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q in L, s(P,Q) = 0. What is the maximum cardinality possible for L?

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