vector inner product

Topics

• linear algebra

For a pair of vectors $x,y\in \mathbb{V}$, the inner product is a scalar function denoted by $⟨x,y⟩:\mathbb{V}\times \mathbb{V}\to \mathbb{F}$, satisfying the following properties:

• Conjugate symmetry: $\forall x,y\in \mathbb{V}:⟨x,y⟩=\overline{⟨y,x⟩}$.
• Linearity: $\forall x_1,x_2,y\in \mathbb{V}$ and $\forall \alpha ,\beta \in \mathbb{F}:⟨\alpha x_1+\beta x_2,y⟩=\alpha ⟨x_1,y⟩+\beta ⟨x_2,y⟩$
• Positive Definiteness: $\forall x\in \mathbb{V}:⟨x,x⟩\ge 0$ and $⟨x,x⟩=0$ if and only if $x=0$

A vector space $\mathbb{V}$ equipped with an inner product $⟨x,y⟩:\mathbb{V}\times \mathbb{V}\to \mathbb{F}$ is called an inner product space.

Examples

• Standard inner product: $x,y\in {\mathbb{R}}^n:⟨x,y⟩=x^{T}y={\sum }_{i=1}^n{x}_i{y}_i$.
• Also, geometrically, $⟨x,y⟩=\Vert x\Vert \Vert y\Vert \cos \theta$ where $\theta$ is the angle between $x$ and $y$.

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