Zero in inner product

Dependencies:

1. Inner product space

$\langle \mathbf{0}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{0} \rangle = 0$.

Proof

\[ \langle \mathbf{0}, \mathbf{v} \rangle = \langle \mathbf{0} + \mathbf{0}, \mathbf{v} \rangle = \langle \mathbf{0}, \mathbf{v} \rangle + \langle \mathbf{0}, \mathbf{v} \rangle \tag{linearity in first argument} \Rightarrow \langle \mathbf{0}, \mathbf{v} \rangle = 0 \]

\[ \langle \mathbf{v}, \mathbf{0} \rangle = \overline{\langle \mathbf{0}, \mathbf{v} \rangle} \tag{conjugate symmetry} = \overline{0} = 0 \]

Dependency for:

1. x and y are parallel iff ∥x∥²∥y∥² = |< x, y >|².
2. Cauchy-Schwarz Inequality

Info:

• Depth: 5
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Inner product space