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. In mathematical notation, a linear combination of vectors $\{\overrightarrow{v_1},\overrightarrow{v_2},\dots ,\overrightarrow{v_n}\}$ with coefficients $\{a_1,a_2,\dots ,a_n\}$ is expressed as:

$$\vec{c}=a_1\overrightarrow{v_1}+a_2\overrightarrow{v_2}+\dots +a_n\overrightarrow{v_n}$$

where $c$ is the resulting vector from the linear combination.

Alternate Definition

$\vec{v}$ is a linear combination of the vectors $\overrightarrow{v_1},\overrightarrow{v_2},\dots ,\overrightarrow{v_n}$ if $\vec{v}$ can be written as:

$$\vec{v}=x_1\overrightarrow{v_1}+x_2\overrightarrow{v_2}+...+x_n\overrightarrow{v_n}$$

where:

• $x_1,x_2,\dots ,x_n\in \mathbb{R}$
• $\overrightarrow{v_1},\overrightarrow{v_2},\dots ,\overrightarrow{v_n}\in V$ where $V$ is a Vector Space