derivation of rank of tensor from the product of two tensors

Question

If $A^p$ is a first rank tensor and $A^pK^{qrs}$ is a 4th rank tensor we have to prove that $K^{qrs}$ is a tensor of rank three?.we can check here clearly that $A^p$ is first rank tensor and $K^{qrs}$ is third rank tensor but i cant understand that why the condition of $A^pK^{qrs}$ is given here??

Answer 1

I hope this works:

As $A^p$ is a first rank tensor we know it belongs to some vector space $V$.

As $A^pK^{qrs}$ is a fourth rank tensor we know it belong to, say, $V\otimes V\otimes V\otimes V$.

The crucial observation is that by universality $V\otimes V\otimes V\otimes V\simeq V \otimes (V\otimes V\otimes V)$, so that you may decompose your fourth rank tensor as $A^p\otimes K^{qrs}$ with $A^p\in V$ and $K^{qrs}\in V\otimes V\otimes V$, hence $K^{qrs}$ is a third rank tensor.