For example, I know that a vector is represented by $a^iv\_i$, where $a^i$ is a component, and $v\_i$ is a covariant basis vector. A dual vector can be represented by $a\_iv^i$, where $a\_i$ is a component and $v^i$ is a contravariant basis vector; but what is represented by ${a\_i}{v\_i}$? Is that just a misuse of notation that early calculus classes use?