Archive for October, 2010

The three musketeers

Given vectors \mathbf{x} and \mathbf{y}, the three quantities \mathbf{x}\cdot\mathbf{x}, \mathbf{x}\cdot\mathbf{y} and \mathbf{y}\cdot\mathbf{y} are very interesting indeed.

In fact, the concept of inner product vector space revolves largely around those quantities. The underlying mathematical fact is the Cauchy-Schwarz inequality (\mathbf{x}\cdot\mathbf{y})^2 \le  (\mathbf{x}\cdot\mathbf{x})\times (\mathbf{y}\cdot\mathbf{y}), which is a corollary of the axioms of an inner product vector space.

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