- #1
Is U+W the Same as U∪W in Vector Spaces?
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Discussion Overview
The discussion centers on the definitions and meanings of the notations U + W and U ∪ W in the context of vector spaces. Participants explore how these terms are used differently in mathematical literature, particularly regarding their implications for subspaces.
Discussion Character
- Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- One participant expresses familiarity with U ∪ W but seeks clarification on the definition of U + W.
- Another participant notes that the meaning of "+" can vary depending on context, suggesting that in set theory, it might denote a different operation than in vector spaces.
- A participant clarifies that in the context of vector spaces, U + W refers to the set of all vectors that can be formed by adding vectors from U and W, specifically h = u + w where u is in U and w is in W.
- It is mentioned that U + W is a subspace of a vector space V, while U ∪ W typically is not a subspace.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the definitions, as there are differing interpretations of the "+" notation and its implications in various contexts.
Contextual Notes
The discussion highlights the dependence on specific definitions provided in different mathematical texts, indicating that the meaning of "+" is not universally agreed upon.
- #2
CompuChip
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I know what U \cup W is, but how is U + W defined?
- #3
- #4
Stephen Tashi
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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 A + B to denote (A \cup B) - (A \cap B).)
The page is talking about vector spaces. In that contex, I think U + W means the vector space consisting of all vectors h that can be expressed as h = u + w where u \in U and w \in W.
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 U + W means the vector space consisting of all vectors h that can be expressed as h = u + w where u \in U and w \in W.
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.
- #5
Deveno
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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.
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