Surjectivity and injectivity of tensor products

Question

Let $\phi_j:V_j\rightarrow W_j$ be linear maps between $\mathbb{R}$-vector spaces for $j=1,...,k$ and define $\psi:=\phi_1 \otimes ... \otimes \phi_k : V_1 \otimes ... \otimes V_k \rightarrow W_1\otimes ... \otimes W_k$.

I need some help to show that the injectivity and surjectivity of the the tensor product $\psi$ does not imply the injectivity and surjectivity of $\phi_1,...,\phi_k$.

Thanks for your help.

Answer 1

For a simple example, if $V_1 = 0 = W_1$, then $V_1\otimes\cdots\otimes V_k \cong 0 \cong W_1\otimes\cdots W_k$. It follows that $\psi$ must be an isomorphism no matter what $\phi_2,\ldots,\phi_k$ are.