Linear combination - Knowlpedia

Let $X$ be a vector space over a field $K$ . Given vectors $x_1,\dots,x_m\in X$ and scalars $\alpha_1,\dots,\alpha_m\in K$ , a linear combination of $x_1,\dots,x_m$ is any vector of the form

\alpha_1x_1+\cdots+\alpha_m x_m=\sum_{i=1}^m \alpha_i x_i.

Only finite sums are allowed in this definition.

Relation to spanning

Linear combinations are the basic algebraic operation behind the span, and a basis is precisely a set that generates every vector via uniquely determined finitely supported coefficients.

Examples

In $\mathbb{R}^2$ , the vector $(3,1)$ is a linear combination of $(1,0)$ and $(0,1)$ via $(3,1)=3(1,0)+1(0,1)$ .
In the polynomial space $P$ , the polynomial $2+5t-t^3$ is a linear combination of $1,t,t^3$ with coefficients $2,5,-1$ .
If $f,g\in F(\Omega)$ , then $\alpha f+\beta g$ is the function $x\mapsto \alpha f(x)+\beta g(x)$ .