Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 6 / v / Solution

Let $B$ be the rank-two tensor and $v$ the vector. The components of their tensor product are $T_{ijk}=B_{ij}{v}_k$. Applying the transformation rules givesThe product therefore obeys the three-index tensor component transformation law, proving that $B_{ij}{v}_k$ is a rank-three tensor. Tensor-product order adds because no pair of indices is summed out.