Linear combination

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. Let $v_1, \dots, v_k \in V$. A linear combination of $v_1, \dots, v_k$ is a vector of the form $$a_1 v_1 + a_2 v_2 + \cdots + a_k v_k,$$ where $a_1, \dots, a_k \in F$ are the coefficients. For an arbitrary family $(v_i)_{i \in I}$, a linear combination is a sum $\sum_{i \in I} a_i v_i$ in which only finitely many coefficients $a_i$ are nonzero.

Remarks

The empty linear combination, with $k = 0$, is the zero vector. The vector $(3, 5)$ is the linear combination $3 e_1 + 5 e_2$ of the standard basis vectors of $F^2$. The set of all linear combinations of a family is its linear span. The matrix–vector product $Ax$ is a linear combination of the columns of $A$, with the entries of $x$ as coefficients.

Explanations

L1 human

A linear combination mixes some vectors: scale each one by a number, then add the results. With the vectors $(1, 0)$ and $(0, 1)$ we can make every vector of the plane in this way. Infinite sums are not allowed: only finitely many coefficients can be nonzero.