Tensor product/Modulo tensor product/Kernel/Exercise

Let $U_1\subseteq V_1,U_2\subseteq V_2,\dots ,U_n\subseteq V_n$ be linear subspaces with the residue class spaces $V_1/U_1,\dots ,V_n/U_n$. Show that the kernel of the canonical mapping

$V_1{\otimes }_{}\cdots {\otimes }_{}{V}_n⟶(V_1/U_1){\otimes }_{}\cdots {\otimes }_{}(V_n/U_n)$

is

$\sum _j{V}_1{\otimes }_{}\cdots {\otimes }_{}{V}_{j-1}\otimes U_j\otimes V_{j+1}{\otimes }_{}\cdots {\otimes }_{}{V}_n.$

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