Suppose $\{v_1,v_2,.....,v_n\}$ are unit vectors in $\mathbb{R}^n$

Question

Suppose $\{v_1,v_2,.....,v_n\}$ are unit vectors in $\mathbb{R}^n$ such that $||v||^2=\sum _{i=1}^n |<v_i,v>|^2,$ for all $v \in \mathbb{R}^n$

Then decide the correct statements in the following.

1. $v_1,v_2,...v_n$ are mutually orthogonal.

2. $\{v_1,v_2,.....,v_n\}$ is basis of $\mathbb{R}^n$

3. $v_1,v_2,...v_n$ are not mutually orthogonal.

4. At most $n-1$ of the elements in the set $\{v_1,v_2,...v_n\}$ can be orthogonal

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

Answer 1

Take $v=v_i$,

Then $\lVert v_i\rVert^2 = \sum_j \lvert\langle v_i,v_j\rangle\rvert^2$, and subtract the $i$th 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