- #1
In summary, the use of "+" in mathematics can vary depending on the context. In the context of sets, it can represent the operation of taking the union and difference of sets. However, in the context of vector spaces, it often denotes the vector space consisting of all possible combinations of vectors from two subspaces. It is important to check the specific meaning given in a particular book or context.
- #2
CompuChip
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I know what [itex]U \cup W[/itex] is, but how is [itex]U + W[/itex] 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 [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.
- #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.
Related to Is U+W the Same as U∪W in Vector Spaces?
1. What is the difference between a union and a sum in scientific terms?
A union refers to the combination of two or more sets or groups into one larger set or group. In scientific terms, it can also refer to the joining of atoms or molecules to form a compound. On the other hand, a sum refers to the result of adding two or more numbers together. In scientific terms, it can also refer to the total amount of a particular substance in a mixture or solution.
2. Are union and sum interchangeable in all scientific fields?
No, union and sum have different meanings and applications in different scientific fields. For example, in mathematics, they are often used interchangeably, but in chemistry, they have distinct meanings as mentioned above.
3. Can a union and a sum be performed on non-numerical data?
Yes, a union and a sum can be performed on non-numerical data. In mathematics, a union can be performed on any sets, regardless of whether they contain numbers or other types of data. In chemistry, a union can refer to the joining of atoms or molecules, which are not numerical. Similarly, a sum can refer to the total amount of a substance in a mixture, which may not be a numerical value.
4. How are unions and sums used in statistical analysis?
In statistical analysis, unions and sums are used to calculate probabilities and determine the likelihood of certain outcomes. A union is used to calculate the probability of two or more events occurring together, while a sum is used to calculate the total probability of all possible outcomes. Both unions and sums are important tools in statistical analysis for making predictions and drawing conclusions.
5. Can a union and a sum be performed on an infinite number of elements?
Yes, a union and a sum can be performed on an infinite number of elements. In mathematics, unions and sums can be performed on infinite sets, as long as they follow certain rules and conditions. In science, these concepts are used to describe and explain natural phenomena that may involve an infinite number of elements, such as the expansion of the universe or the behavior of subatomic particles.
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