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?
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?
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?
Show Hint
Ortho (90 degrees Zero dot product) + Normal (Length is 1) = Orthonormal.
Updated On: Jul 3, 2026
- both I and II are true
- only I is true
- only II is true
- neither I nor II is true
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Concept
The term "Orthonormal" combines two properties: Orthogonality and Normalization.
Step 2: Meaning
"Orthogonal" means the inner product of any two distinct vectors is zero ($(\alpha_i, \alpha_j) = 0$ for $i \neq j$), which is statement II.
Step 3: Analysis
"Normal" means each vector is a unit vector, having a norm of 1 ($||\alpha_i|| = 1$), which is statement I.
Step 4: Conclusion
Since both properties must be present for a set to be orthonormal, both statements I and II are true. Final Answer: (A)
The term "Orthonormal" combines two properties: Orthogonality and Normalization.
Step 2: Meaning
"Orthogonal" means the inner product of any two distinct vectors is zero ($(\alpha_i, \alpha_j) = 0$ for $i \neq j$), which is statement II.
Step 3: Analysis
"Normal" means each vector is a unit vector, having a norm of 1 ($||\alpha_i|| = 1$), which is statement I.
Step 4: Conclusion
Since both properties must be present for a set to be orthonormal, both statements I and II are true. Final Answer: (A)
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