Suppose {v1,v2,…,vn}\{v_1,v_2,…,v_n\} are unit vectors in Rn\mathbb{R}^n

Suppose {v1,v2,…..,vn}\{v_1,v_2,…..,v_n\} are unit vectors in Rn\mathbb{R}^n such that ||v||2=∑ni=1||2,||v||^2=\sum _{i=1}^n ||^2, for all v∈Rnv \in \mathbb{R}^n

Then decide the correct statements in the following.

v1,v2,…vnv_1,v_2,…v_n are mutually orthogonal.
{v1,v2,…..,vn}\{v_1,v_2,…..,v_n\} is basis of Rn\mathbb{R}^n
v1,v2,…vnv_1,v_2,…v_n are not mutually orthogonal.
At most n−1n-1 of the elements in the set {v1,v2,…vn}\{v_1,v_2,…v_n\} can be orthogonal

Since the given vectors are unit vectors, 1 and 2 are right and any n−1n-1 unit vectors are not orthogonal. Is I am right?

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You are correct that 11 and 22 are correct. But can you provide stronger reasoning?
– Bye_World
2 days ago

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1 Answer
1

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Take v=viv=v_i,

Then ‖vi‖2=∑j|⟨vi,vj⟩|2\lVert v_i\rVert^2 = \sum_j \lvert\langle v_i,v_j\rangle\rvert^2, and subtract the iith term from both sides. If a sum of nonnegative terms is zero, then each term is zero, so you get that 1 is true. Hence 2 as well. Not 3 or 4