Vector addition and scalar multiplication can be combined into a single construction. Given vectors $\mathbf v_1,\ldots,\mathbf v_k$, any expression of the form

$$c_1\mathbf v_1+\cdots+c_k\mathbf v_k$$

is a linear combination of those vectors.

The numbers $c_1,\ldots,c_k$ are the coefficients of the combination.

Building one vector from others

For example, let

$$\mathbf e_1=(1,0),\qquad \mathbf e_2=(0,1).$$

Then

$$(3,-2)=3\mathbf e_1-2\mathbf e_2.$$

So the vector $(3,-2)$ has been built from the two coordinate directions.

Solving for coefficients

To ask whether a vector $\mathbf b$ can be built from given vectors is to ask whether there are coefficients satisfying

$$c_1\mathbf v_1+\cdots+c_k\mathbf v_k=\mathbf b.$$

Equating components turns this vector equation into ordinary scalar equations.

Linear combinations are the basic mechanism behind span, linear independence, bases and linear systems.

Visual intuition: linear combinations