Pistetulo

Pistetulo, also known as the dot product, is a binary operation that takes two vectors of equal length and returns a scalar. For real vectors a = (a1, ..., an) and b = (b1, ..., bn), pistetulo is a1 b1 + a2 b2 + ... + an bn. In common mathematical practice for complex vectors, the standard convention uses complex conjugation on one component, so the inner product is a · b = ∑ ai conjugate(bi).

Geometrically, the pistetulo relates to the lengths of the vectors and the angle between them: a · b = ||a|| ||b|| cos θ, where θ is the angle between a and b. This leads to important consequences, such as a · b = 0 if and only if a and b are orthogonal (perpendicular). The pistetulo also satisfies a · a = ||a||^2, connecting the operation to vector lengths.

Algebraically, pistetulo is bilinear and symmetric in real vector spaces, meaning it distributes over vector addition and scalar multiplication, and a · b = b · a. In real spaces it is also positive definite: a · a > 0 for all nonzero a. It underpins many linear-algebra constructions, including projections and orthogonal decompositions (for example, the projection of a onto b is ((a · b)/(b · b)) b) and the Gram-Schmidt process for building orthonormal bases.

Generalization and usage: pistetulo serves as the standard inner product on Euclidean space and as the starting point for inner-product spaces in linear algebra. It is widely used in physics, computer graphics, statistics, and machine learning to measure similarity, angles, and projections between vectors.