Question

Let \((V, \langle,\rangle)\) be an inner product space and let \(\langle x,y\rangle = \|x\|\|y\|\) for all \(x,y \in V\). Then which one of the following options is true?

• A. \(\{x,y\}\) is a linearly independent set.
• B. \(\{x,y\}\) is a linearly dependent set.
• C. \(\{x,y\}\) is an orthogonal set.
• D. \(\{x,y\}\) is an orthonormal set.

Hint:

Compare the given condition with the equality case of the Cauchy-Schwarz inequality.

Answer 1

The Correct Option is B

Step 1: Recall the Cauchy-Schwarz inequality: for any \(x,y\) in an inner product space, \(|\langle x,y\rangle| \le \|x\|\|y\|\), with equality holding if and only if \(x\) and \(y\) are linearly dependent.

Step 2: We are given that \(\langle x,y\rangle = \|x\|\|y\|\) for all \(x,y \in V\). Since \(\|x\|\|y\| \ge 0\), this value equals \(|\langle x,y\rangle|\), which is exactly the equality case of the Cauchy-Schwarz inequality.

Step 3: By the equality condition of Cauchy-Schwarz, \(x\) and \(y\) must be linearly dependent, that is, one is a scalar multiple of the other.

Step 4: Since this holds for every pair \(x,y \in V\), any two vectors chosen from \(V\) are linearly dependent, so \(\{x,y\}\) is a linearly dependent set. This also rules out orthogonality and orthonormality, since two nonzero dependent vectors cannot be orthogonal.

\[\boxed{\{x,y\} \text{ is a linearly dependent set}}\]