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I came across a notation: $\mathbf{v} + U$ where $\mathbf{v}$ is a vector in $V$ and $U$ is a subspace of $V$. What does this notation mean? I've only seen sum of vector spaces but not vector + vector space? Thanks!

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It depends on the context, but sometimes $\vec v+U$ refers to the set

$$\vec v + U := \{\vec v+\vec u:\space \vec u \in U\}$$

Note that this set is not necessarily a subspace if $U$ is a subspace.

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