Innenprodukt
The inner product, also known as the dot product, is a fundamental concept in linear algebra and vector calculus. It is an operation that takes two vectors and returns a scalar. The inner product of two vectors a and b, denoted as a · b, is defined as the sum of the products of their corresponding components. For vectors in Euclidean space, this can be expressed as:
a · b = a1b1 + a2b2 + ... + anbn
where a = (a1, a2, ..., an) and b = (b1, b2, ..., bn) are the vectors.
The inner product has several important properties. It is commutative, meaning a · b = b · a, and
The inner product is closely related to the concept of projection. The projection of vector a onto
proj_b(a) = (a · b / |b|^2) * b
The inner product is also used to define the angle between two vectors. The cosine of the
In the context of matrices, the inner product can be generalized to the matrix product, which is