inner products
An inner product is a mathematical operation that takes two vectors and produces a scalar (a single number). It measures how much one vector extends in the direction of another, providing a way to determine angles and lengths in vector spaces. Common examples of inner products include the dot product in Euclidean space and the complex inner product in complex vector spaces.
Inner products have important properties, such as being linear and symmetric. They also allow for the definition of concepts like orthogonality, where two vectors are perpendicular if their inner product is zero. This makes inner products essential in fields like linear algebra, physics, and machine learning.