Inner product (linear algebra)

Inner product (linear algebra) . In mathematics, the inner product, also known as the dot product, inner product, or dot product, is an application whose domain is V 2 and its co-domain is K, where V is a vector space and K is the respective set of scalars. It is an application that corresponds to two elements of a linear space an element of the body of scalars.

Summary

[ hide ]

• 1 Real case
• 2 Complex case
• 3 Examples
• 4 Schwarz’s theorem
• 5 Sources

Real case

Let V be a real linear space. An internal product in V is a function that assigns to each ordered pair of vectors u and v in V, a real number that is denoted < u , v > and verifies the following formalities:

• < u, v > = < v , u > for all u and v in V ,.
• <α u+ β v , w > = α < u , w > + β < v , w >, in addition < w , α u + β v > = α < w , u > + β < w , v > for all u , v and w in V, α and β real numbers.
• < u, u > positive if v ≠ 0 . Also < u , u > = 0 s, s, s u = 0 .
• The angle φ between two different vectors of 0, u and v is defined by its cosine:

cosφ = < u , v > / < u , u > ^1/2 < v , v > ^1/2

Complex case

Let V be a complex linear space. An internal product in V is a function that assigns to each ordered pair of vectors u and v ‘ in V, a complex number, possibly denoting < u , v > and verifying the following conditions:

• < u ‘ , v > = <v , u > * for all u and v in V, where * designates the conjugate of the result in complex number.
• <α u ‘ + βv , w > = α < u , w > + β < v , w >, in addition < w , α u + β v > = α < w , u > + β < w , v > for all u , v and w in V, α and β real numbers.
• < u, u > positive if v ≠ 0. Also < u , u > = 0 s, s, s u = 0.

Examples

1. Typical R is known as internal product ⁿto the actual number defined as < u , v > = u ₁ v ₁ + … + u _n v _n
2. It is considered as a typical internal productin C ⁿ defined as < u , v > = u ^* ₁ v ₁ + … + u ^* _n v _n
3. < u, v > equal to the integral defined between 0 and 1 of the product of the functions u (t) and v (t) in the real space of the continuous domain functions [0, 1].

Schwarz’s theorem

In mathematics, and more specifically in differential calculus, the Clairaut theorem, also known as the Schwarz theorem or the cross-derivative equality theorem, is a sufficient condition for the equality of the crossed partial derivatives of a function of several variables. The theorem states that if the crossed partial derivatives exist and are continuous, then they are equal.

Given an interior product in the real or complex vector space V, it is stated that

<uv> ≤ <u, u> ^1/2 <v, v> ^1/2