Question:
In an inner product vector space if vectors \(\alpha,\beta\) are linearly dependent vectors then
In an inner product vector space if vectors \(\alpha,\beta\) are linearly dependent vectors then
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Equality in Cauchy-Schwarz inequality holds exactly when the vectors are linearly dependent.
Updated On: Jul 3, 2026
- \(|(\alpha,\beta)|\leq \|\alpha\|\|\beta\|\)
- \(|(\alpha,\beta)|\geq \|\alpha\|\|\beta\|\)
- \(|(\alpha,\beta)|< \|\alpha\|\|\beta\|\)
- \(|(\alpha,\beta)|=\|\alpha\|\|\beta\|\)
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The Correct Option is D
Solution and Explanation
Concept:
The Cauchy-Schwarz inequality says \[ |(\alpha,\beta)|\leq \|\alpha\|\|\beta\| \] Equality holds if and only if \(\alpha\) and \(\beta\) are linearly dependent.
Step 1: Given condition. The vectors \[ \alpha,\quad \beta \] are linearly dependent. Therefore, one vector is a scalar multiple of the other. \[ \beta=k\alpha \]
Step 2: Use equality condition of Cauchy-Schwarz. For linearly dependent vectors, \[ |(\alpha,\beta)|=\|\alpha\|\|\beta\| \]
Step 3: Final answer. \[ \boxed{|(\alpha,\beta)|=\|\alpha\|\|\beta\|} \]
The Cauchy-Schwarz inequality says \[ |(\alpha,\beta)|\leq \|\alpha\|\|\beta\| \] Equality holds if and only if \(\alpha\) and \(\beta\) are linearly dependent.
Step 1: Given condition. The vectors \[ \alpha,\quad \beta \] are linearly dependent. Therefore, one vector is a scalar multiple of the other. \[ \beta=k\alpha \]
Step 2: Use equality condition of Cauchy-Schwarz. For linearly dependent vectors, \[ |(\alpha,\beta)|=\|\alpha\|\|\beta\| \]
Step 3: Final answer. \[ \boxed{|(\alpha,\beta)|=\|\alpha\|\|\beta\|} \]
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