What is inner product?

The inner product is an operation between two vectors that results in a scalar value. Given two vectors, $u$ and $v$, the inner product is $\langle u,v \rangle$.

Examples

• The inner product between two real numbers is the product of numbers.

• In Euclidean space, the inner product is the dot product between vectors.

• In the vector space of real functions on a closed interval $[a, \space b]$, the inner product is the integral of the product of the two functions over the interval.

Properties of inner product

Let $u$, $v$ and $w$ be vectors and $c$ be a scalar.

• Symmetry: $\langle u, v \rangle$ = $\langle v, u \rangle$

• Distributivity: $\langle u+v, w \rangle$= $\langle u, w \rangle$+ $\langle v, w \rangle$

• Homogenity: $\langle cu, v \rangle$= $c \space\langle u, v \rangle$

• Positivity: $\langle v, v \rangle \space \geq \space 0$, where $\langle v, v \rangle \space = \space 0$ if and only if $v$ is a zero vector.

Inner product space

An inner product space is a vector space $V$ with an inner product on $V$.

The inner product defines the following concepts in an inner product space.

Norm

The norm is the square root of the inner product of the vector with itself.

Distance

The distance is the norm of the difference between the vectors.

Angle between vectors

The angle between vectors is the inverse cosine of the normalized inner product.

Orthogonal vectors

Two vectors are said to be orthogonal if their inner product is zero, i.e.,

Geometrically, orthogonal vectors are perpendicular to each other.

Conclusion

In conclusion, the inner product is an operation that calculates a scalar value from two vectors, providing essential properties and applications in various mathematical contexts such as vector spaces and inner product spaces.