Linear Combinations

A sum of scalar multiples of vectors in a list is called a linear combination of the list.

Definition: Linear Combination

A linear combination of a list $v_1,\dots ,v_m$ of vectors in $V$ is a vector of the form

$$a_1{v}_1+\cdots +a_m{v}_m$$

where $a_1,\dots ,a_m\in \mathbb{F}$.

For example, $(17,-4,2)$ is a linear combination of $(2,1,-3),(1,-2,4)$, which is a list of length two of vectors in ${\mathbb{R}}^3$, because

$$(17,-4,2)=6(2,1,3)+5(1,-2,4)$$

On the other hand, $(17,-4,5)$ is NOT a linear combination of $(2,1,-3),(1,-2,4)$, because there do not exist numbers $a_1,a_2\in \mathbb{F}$ such that

$$(17,-4,5)=a_1(2,1,-3)+a_2(1,-2,4)$$