Linear Combinations

A linear combination of vectors $\mathbf{a}_1, \ldots, \mathbf{a}_k$ is any expression $c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 + \cdots + c_k \mathbf{a}_k$ where $c_1, \ldots, c_k$ are scalars. The scalars are called the coefficients or weights.

Linear combinations are the core operation of linear algebra. Matrix-vector multiplication $A\mathbf{x}$ is a linear combination of the columns of $A$ weighted by the entries of $\mathbf{x}$.

There are three ways to picture a linear combination: symbolically (the formula above), geometrically (tip-to-tail addition of scaled arrows), and entry-by-entry (compute each entry of the result separately). All three perspectives are useful.