User:Pcasau

Inner Product

Let $\mathbb{F}^n$ denote the $n$-dimensional Vector Space over the field of complex numbers $\mathbb{F}=\mathbb{C}$ or real numbers $\mathbb{F}=\mathbb{R}$ Let $x,y\in \mathbb{F}^n$ be given by \begin{align*} x&:=\begin{bmatrix} x_1\\ \vdots\\ x_n \end{bmatrix} & y&:=\begin{bmatrix} y_1\\ \vdots\\ y_n \end{bmatrix} \end{align*} The inner product between $x$ and $y$ is given by $$x^* y:=\sum_{i=1}^n \bar{x}_i y_i.$$

The inner product between a vector and itself is zero if and only if the vector is the zero vector

Given $x\in\mathbb{F}^n$

$$x^* x=0 \iff x = 0$$

Proof

From Inner Product $$x^* x = \sum_{i=1}^n \bar{x}_i x_i$$ From Product of Complex Number with Conjugate and Definition:Complex Modulus $$x^* x = \sum_{i=1}^n |x_i|^2$$ From Complex Modulus is Norm $$x^* x=0 \iff x = 0$$