Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Algebraic Properties: Dot Product of Vectors

Let $\vec{x}, \vec{y}, \vec{z} \in R^{n_{x} 1}$ and $a, b \in R$ . Then, all of the following are algebraic properties of the 5.1 Inner Product Between Vectors.

Bilinearity:

Linearity in left argument: $(a \vec{x} + b \vec{y}) \cdot \vec{z} = a \cdot (\vec{x} \cdot \vec{z}) + b \cdot (\vec{y} \cdot \vec{z})$
Linearity in right argument: $\vec{x} \cdot (a \vec{y} + b \vec{z}) = a \cdot (\vec{x} \cdot \vec{y}) + b \cdot (\vec{x} \cdot \vec{z})$

Symmetry: $\vec{x} \cdot \vec{y} = \vec{y} \cdot \vec{x}$
Positivity: $\vec{x} \cdot \vec{x} > \vec{0}$ when $\vec{x} \neq 0$ while $\vec{0} \cdot \vec{0} = 0$ .

Linearity in Right Argument

\begin{proof}
Let $\vec{x}, \vec{y}, \vec{z} \in R^{n}$ and $a, b \in R$ . Consider:

\begin{aligned} \vec{x} \cdot (a \vec{y} + b \vec{z}) & = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] \cdot (a \cdot [\begin{array}{c} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{array}] + b \cdot [\begin{array}{c} z_{1} \\ z_{2} \\ ⋮ \\ z_{n} \end{array}]) \\ = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] \cdot ([\begin{array}{c} a y_{1} \\ a y_{2} \\ ⋮ \\ a y_{n} \end{array}] + [\begin{array}{c} b z_{1} \\ b z_{2} \\ ⋮ \\ b z_{n} \end{array}]) \\ = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] \cdot [\begin{array}{c} a y_{1} + b z_{1} \\ a y_{2} + b z_{2} \\ ⋮ \\ a y_{n} + b z_{n} \end{array}] \\ = x_{1} \cdot (a y_{1} + b z_{1}) + x_{2} \cdot (a y_{2} + b z_{2}) + \dots + x_{n} (a y_{n} + b z_{n}) \\ = \sum_{j = 1}^{n} x_{i} (a y_{i} + b z_{i}) \\ = (a x_{1} y_{1} + b x_{1} z_{1}) + (a x_{2} y_{2} + b x_{2} z_{2}) + \dots + (a x_{n} y_{n} + b x_{n} z_{n}) \\ = (a x_{1} y_{1} + a x_{2} y_{2} + \dots + a x_{n} y_{n}) + (b x_{1} z_{1} + b x_{1} z_{2} + \dots + b x_{n} z_{n}) \\ = a (x_{1} y_{1} + x_{2} y_{2} + \dots + x_{n} y_{n}) + b (x_{1} z_{1} + x_{2} z_{2} + \dots + x_{n} z_{n}) \\ = a (\vec{x} \cdot \vec{y}) + b (\vec{x} \cdot \vec{z}) \end{aligned}

\end{proof}