$(\mathrm{Vect}_k,\otimes_k)$ as a non-strict monoidal category

Question

What is the easiest way to see (or understand) that the category of vector spaces over a field $k$, endowed with its usual modoidal structure $\otimes_k$, is not a strict monoidal category.

Answer 1

$k\otimes V$ is a certain quotient of the vector space freely generated by $k\times V$, so its elements are equivalence classes of formal sums of pairs $(x,v),x\in k,v\in V$. There's no way such a thing is equal to an element of $V$, so $k\otimes V$ isn't equal to $V$.