Let $V$ and $W$ be two vector spaces over $\mathbb{F}$. We know that their tensor product $V \otimes W$ is also a vector space over $\mathbb{F}$. I am wondering how would look like the elements inside it? I.e. if I take a vector $v \in V$ and $w \in W$, does $v \otimes w$ behave as the Kronecker product of $v$ and $w$?

If I take for example $\mathbb{C}\otimes_{\mathbb{R}} \mathbb{R}$ for example, what is the explicit form of its elements?


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