= Linear combination
{title2=$\sum_{i=1}^n a_i v_i$}
{wiki}
Let $V$ be a <vector space> over a specified <field> $F$, and let $v_1,\ldots,v_n\in V$. A <linear combination> is a finite sum
$$
w=\sum_{i=1}^n a_i v_i,\qquad a_i\in F.
$$
The <coefficients> $a_i$ are <scalars> from that same <field>; each term uses <scalar multiplication>, and the terms are joined by <vector addition>. For <coefficients> in the <real numbers>, negative values reverse arrows and fractional values scale them by fractional amounts. The collection of attainable <vectors> is the <linear span>. The column picture in Strang's MIT notes interprets solving a <linear system> as choosing <coefficients> to reach a target <vector>.https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/6913aa2f37dada8fdda334168b30339b_MIT18_06SCF11_Ses1.1sum.pdf[MIT 18.06SC, Strang lecture summary 1, pp. 2–3: the column picture]{ref}
For a concrete calculation over the <real numbers>, take
$$
v_1=(1,1),\qquad v_2=(1,-1),\qquad w=(3,1).
$$
Then $2v_1+v_2=(2,2)+(1,-1)=(3,1)=w$. More generally, solving $a+b=x$ and $a-b=y$ gives
$$
(x,y)=\frac{x+y}{2}v_1+\frac{x-y}{2}v_2.
$$
Every <vector> in the <Euclidean plane> is therefore a <linear combination> of $v_1,v_2$, so their <linear span> is all of $\mathbb R^2$. The target $(3,1)$ has <coefficients> $(2,1)$; the target $(0,2)$ has <coefficients> $(1,-1)$, since $(1,1)-(1,-1)=(0,2)$. The displayed formula also makes the <coefficients> unique. Consequently $(v_1,v_2)$ is an <ordered basis>, with <coordinate vector> $[(x,y)]_{(v_1,v_2)}=((x+y)/2,(x-y)/2)^T$.
\Image[/linear-combination-coordinates.png]
{title=A <linear combination> reaches (3,1) with <coefficients> (2,1)}
{description=The shaded <parallelogram> has sides $2v_1$ and $v_2$; either order of <vector addition> reaches $w=(3,1)$. The grid lines hold one <coefficient> fixed while varying the other. They show how the <ordered basis> $v_1=(1,1),v_2=(1,-1)$ supplies a unique <coordinate vector> for each <vector> in the <Euclidean plane>.}
The payoff is a reusable language for synthesis: <linear combinations> build <vectors> from chosen ingredients, <linear span> decides which targets are reachable, and <linear independence> decides whether the <coefficients> are unique. The ingredients can also be <polynomials> or <functions>, as long as they belong to a <vector space> over the specified <field>.
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