Description

Inner products are operations from V×V→RV \times V \rightarrow \mathbb{R}. They must fulfill the following 4 properties for any x⃗,y⃗,z⃗∈V\vec{x}, \vec{y}, \vec{z} \in V and α∈R\alpha \in \mathbb{R}:

  1. Commutativity: ⟨x⃗,y⃗⟩=⟨y⃗,x⃗⟩\langle \vec{x}, \vec{y}\rangle = \langle \vec{y}, \vec{x}\rangle

  2. Distributivity: ⟨x⃗,y⃗+z⃗⟩=⟨x⃗,y⃗⟩+⟨x⃗,z⃗⟩\langle \vec{x}, \vec{y} + \vec{z}\rangle = \langle \vec{x}, \vec{y}\rangle + \langle \vec{x}, \vec{z}\rangle

  3. Scaling: ⟨αx⃗,y⃗⟩=α⟨x⃗,y⃗⟩\langle \alpha\vec{x}, \vec{y}\rangle = \alpha\langle \vec{x}, \vec{y}\rangle

  4. Nonnegativity: ⟨x⃗,x⃗⟩≥0\langle \vec{x}, \vec{x}\rangle \geq 0, where ⟨x⃗,x⃗⟩=0\langle \vec{x}, \vec{x}\rangle = 0 if and only if x⃗=0\vec{x} = 0.

In this class, we are going to use the inner product in the vector space Rn\mathbb{R}^n that is defined as follows (commonly known as the dot product): Let x⃗=[x1x2⋮xn]\vec{x} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} and y⃗=[y1y2⋮yn]\vec{y} = \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}.

⟨x⃗,y⃗⟩=⟨[x1x2⋮xn],[y1y2⋮yn]⟩=x1y1+x2y2+⋯+xnyn\langle \vec{x}, \vec{y}\rangle = \left\langle \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix}, \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}\right\rangle = x_1y_1 + x_2y_2 + \cdots + x_ny_n

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