Linear and multilinear algebra; matrix theory Definition

Dot product

Definition

Let $n \ge 1$ and let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ be vectors of $\R^n$ . The dot product of $x$ and $y$ is the real number $x \cdot y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n.$

Remarks

In matrix notation, $x \cdot y = x^T y$ for column vectors $x$ and $y$ . It is symmetric, bilinear and positive definite, so it is an inner product on $\R^n$ . Geometrically, $x \cdot y = \norm{x} \norm{y} \cos\theta$ , where $\theta$ is the angle between $x$ and $y$ . The same formula defines a symmetric bilinear form on $F^n$ for every field $F$ .

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Dot product definition

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Tag	`05HQYM0R4G`
URI	`m2://05HQYM0R4G`
Kind	definition
Writing	W1 stated
MSC	15A63
Defines	Dot product
Uses	Coordinate space

Dot product

Definition

Remarks

Reading path

Where this leads

Further reading

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