linear combination

A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.

Formally,
Let $V$ be a vector space over the field $K$ . The elements of $V$ are vectors, and the elements of $K$ are scalars.
A linear combination of vectors ${\vec{v}}_{i}$ using scalars $a_{i}$ as coefficients takes the form

a_{1} {\vec{v}}_{1} + a_{2} {\vec{v}}_{2} + a_{3} {\vec{v}}_{3} + \dots + a_{n} {\vec{v}}_{n}

The set of all linear combinations of ${\vec{v}}_{1}, \dots, {\vec{v}}_{n}$ always forms a subspace. This set is called the span of the vectors.