Linear combination

Definition

Let $V$ be a vector space over the field $F$. Let $S$ be a nonempty subset of $V$. A nonempty vector $\mathbf{v}\in V$ is called a linear combination of vectors of $S$ if there exist a finite number of vectors ${\mathbf{u}}_1,\cdots ,{\mathbf{u}}_n$ and scalars $a_i,\cdots ,a_n$ such that

$\mathbf{v}=\sum \_{i=1}^{n}a\_i\mathbf{u}\_i.$

Then, $\mathbf{v}$ is called a linear combination of vectors $\mathbf{u}\_1,\cdots ,\mathbf{u}\_n$, and call $a\_i,\cdots ,a\_n$ the coefficients of the linear combination.

Observations

Observation 1

In any vector space $V$, $0\mathbf{v}=0$ for any $\mathbf{v}\in V$.

Thus the zero vector is a linear combination of any nonempty subset of $V$.

Related Concepts

• Span: The set of all linear combinations of some set of vectors