lineaarikombinaatiolla

Lineaarikombinaatiolla, or linear combination in English, is a fundamental concept in linear algebra. It refers to an expression constructed from a set of vectors by multiplying each vector by a scalar (a number) and then adding the results. Given a set of vectors v₁, v₂, ..., vₙ and a set of scalars c₁, c₂, ..., cₙ, their linear combination is represented as c₁v₁ + c₂v₂ + ... + cₙvₙ.

The resulting vector from a linear combination lies within the span of the original vectors. The span of a set of vectors is the set of all possible linear combinations that can be formed from those vectors. Understanding linear combinations is crucial for grasping concepts like vector spaces, basis, and dimension. They are used extensively in various fields, including physics, engineering, computer graphics, and data science, for modeling and solving problems involving multiple variables and relationships. The coefficients, or scalars, in a linear combination determine the contribution of each vector to the final result.