linear combination

Given vectors \mathbf v_1,\dots,\mathbf v_k and scalars c_1,\dots,c_k, a linear combination of the \mathbf v_i is the vector c_1\mathbf v_1 + c_2\mathbf v_2 + \cdots + c_k\mathbf v_k, with each c_i called the coefficient of \mathbf v_i. There is no restriction on the scalars — positive, negative, zero, or fractional — and no restriction on k. Setting every coefficient to 0 always gives \mathbf 0; setting one coefficient to 1 and the rest to 0 returns that single vector unchanged, so a linear combination generalizes both scalar multiplication and addition rather than competing with them.