Dot Product is Inner Product

From ProofWiki
Jump to navigation Jump to search

Theorem

The dot product is an inner product.


Proof

Let $\mathbf u, \mathbf v \in \R^n$.

We will check the four defining properties of an inner product in turn.


Conjugate Symmetry

\(\displaystyle \mathbf u \cdot \mathbf v\) \(=\) \(\displaystyle \mathbf v \cdot \mathbf u\) Dot Product Operator is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \overline{\mathbf v \cdot \mathbf u}\) Complex Number equals Conjugate iff Wholly Real

Thus the dot product possesses conjugate symmetry.

$\Box$


Bilinearity

From Dot Product Operator is Bilinear, the dot product possesses bilinearity.

$\Box$


Non-Negative Definiteness

From Dot Product with Self is Non-Negative, the dot product possesses non-negative definiteness.

$\Box$


Positiveness

From Dot Product with Self is Zero iff Zero Vector, the dot product possesses positiveness.

$\Box$


Hence the dot product satisfies the definition of an inner product.

$\blacksquare$