lineaarkombinatsiooni

Lineaarkombinatsiooni, often translated as 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 and then summing the results. Mathematically, if we have a set of vectors v1, v2, ..., vn and a corresponding set of scalars c1, c2, ..., cn, their linear combination is given by c1*v1 + c2*v2 + ... + cn*vn.

The resulting vector from this operation lies within the span of the original vectors. The concept of linear combination is crucial for understanding vector spaces, linear independence, basis, and the solution spaces of linear systems. For instance, a solution to a homogeneous system of linear equations can be expressed as a linear combination of the basis vectors of its null space. Similarly, any vector in a vector space can be represented as a linear combination of its basis vectors. The coefficients in this combination are unique if the set of vectors forms a basis. The ability to form linear combinations is what defines the structure and properties of vector spaces.