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**arbolis** What's the difference between $\displaystyle V \cup W$ and $\displaystyle V \oplus W$, $\displaystyle V$ and $\displaystyle W$ are vector spaces. I didn't find the answer on the Internet.

A friend told me that $\displaystyle V+W=$$\displaystyle \text {span} (V \cup W)$. But then if $\displaystyle V$ and $\displaystyle W$ are $\displaystyle \mathbb{R}^2$ as vector space over $\displaystyle \mathbb{R}$ then I get that $\displaystyle \mathbb{R}^2+\mathbb{R}^2$ is formed by ... ahh well I'm unsure of everything.