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Is U+W the Same as U∪W in Vector Spaces?
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I know what [itex]U \cup W[/itex] is, but how is [itex]U + W[/itex] defined?
athrun200
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There many different meanings for mathematical notations, depending on the context. The page you gave is not talking about "+" in the context of sets. (In the context of sets, some books use [itex]A + B[/itex] to denote [itex](A \cup B) - (A \cap B)[/itex].)
The page is talking about vector spaces. In that contex, I think [itex]U + W[/itex] means the vector space consisting of all vectors [itex]h[/itex] that can be expressed as [itex]h = u + w[/itex] where [itex]u \in U[/itex] and [itex]w \in W[/itex].
However, if you want to be sure of the meaning of "+" in a particular book, you must see what that book says it means. There is no "universal" meaning for it.
The page is talking about vector spaces. In that contex, I think [itex]U + W[/itex] means the vector space consisting of all vectors [itex]h[/itex] that can be expressed as [itex]h = u + w[/itex] where [itex]u \in U[/itex] and [itex]w \in W[/itex].
However, if you want to be sure of the meaning of "+" in a particular book, you must see what that book says it means. There is no "universal" meaning for it.
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for vector spaces, particularly when U,W are subspaces of a vector space V,
U+W is a subspace of V, U∪W usually is not.
U+W is a subspace of V, U∪W usually is not.