Question

What is a linear combination of vectors?

Answer 1

If $\mathbf{w}$ is a vector in $R^{n}$, then $\mathbf{w}$ is said to be a linear combination of the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{r}$ in $R^{n}$ if it can be expressed in the form $$ \mathbf{w}=k_{1} \mathbf{v}_{1}+k_{2} \mathbf{v}_{2}+\cdots+k_{r} \mathbf{v}_{r} $$ where $k_{1}, k_{2}, \ldots, k_{r}$ are scalars. These scalars are called the coefficients of the linear combination. In the case where $r=1$, Formula (14) becomes $\mathbf{w}=k_{1} \mathbf{v}_{1}$, so that a linear combination of a single vector is just a scalar multiple of that vector.