Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Linear Algebra Inner Product Spaces, Orthonormal Basis
Let ⟨·, ·⟩ denote the standard inner product on ℝ7. Let Σ = {v1, . . ., v5} ⊆ ℝ7 be a set of unit vectors such that ⟨vi, vj⟩ is a non-positive integer for all 1 ≤ i ≠ j ≤ 5. Define N(Σ) to be the number of pairs (r, s), 1 ≤ r, s ≤ 5, such that ⟨vr, vs⟩ ≠ 0. The maximum possible value of N(Σ) is equal to
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