Question

Let V be an inner product space and $S = \{\alpha_{1}, \alpha_{2}, \dots, \alpha_{m}\}$ be a finite subset of V. If S is an orthonormal set, then consider the following statements:
I: $||\alpha_{i}|| = 1$ for each $\alpha_{i} \in S$
II: $(\alpha_{i}, \alpha_{j}) = 0$ for $\alpha_{i}, \alpha_{j} \in S, i \neq j$.
Which of the following is correct?

• A. both I and II are true
• B. only I is true
• C. only II is true
• D. neither I nor II is true

Hint:

Ortho (90 degrees Zero dot product) + Normal (Length is 1) = Orthonormal.

Answer 1

The Correct Option is A