- #1
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SUMMARY
In vector spaces, the notation U + W represents the vector space formed by all vectors h that can be expressed as h = u + w, where u is an element of U and w is an element of W. This operation results in a subspace of a larger vector space V. In contrast, U ∪ W denotes the union of sets U and W, which typically does not form a subspace. The interpretation of the "+" symbol can vary depending on the context, and it is essential to refer to specific texts for precise definitions.
PREREQUISITES- Understanding of vector spaces and subspaces
- Familiarity with set operations, specifically union and intersection
- Knowledge of mathematical notation and its contextual meanings
- Basic linear algebra concepts
- Study the definition and properties of vector spaces in linear algebra
- Learn about the concepts of subspaces and their significance in vector spaces
- Research the differences between set operations and vector space operations
- Examine various mathematical texts to understand notation variations
Students of mathematics, particularly those studying linear algebra, educators teaching vector space concepts, and anyone seeking clarity on mathematical notation in different contexts.
- #2
CompuChip
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I know what U \cup W is, but how is U + W defined?
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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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