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Given two subspaces, $U$ and $W$ of $V$, I know that $U+W$ and $U\cup W$ are related in that $U+W$ is the smallest subspace containing $U\cup W$, but what's their relationship in $U+W, U\cup W$, and $U \bigoplus W$?

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We have $U \cup V \subseteq U+V$ and $span(U \cup V)=U+V$.

$U \oplus W$ is the notation for $U+V,$ if $U \cap V=\{0\}$.

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