Why span(k) Contains More Than k

  • Thread starter nhrock3
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However, I don't see how it is possible for any other subspace W to be a subset of any set of vectors.In summary, the conversation discusses the relationship between two mathematical concepts: k, a group of vectors, and W, a subspace. It is stated that if W is a subset of k, then it is also a subset of the "wider" set span(k), which contains all possible combinations of k. However, there is some confusion about this statement as a subspace typically consists of an infinite number of vectors and could not be a subset of a finite collection of vectors.
  • #1
nhrock3
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k is a group of vectors

W is a subspace.

why if W [tex]\subseteq[/tex]k then W [tex]\subseteq[/tex]span(k)

??

span(k) is wider then k



it contains all the combinations of k
 
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  • #2
You're logic seems fine. Certainly if W is a subset of k, then W is a subset of something 'wider' than k (span{k}).
 
  • #3
There's something that bothers me about this. If W is a subspace, and W is also a subset of a set of vectors, then it seems to me that W must be the zero vector in whatever space we happen to be working with.

Any other subspaces consist of an infinite number of vectors, so couldn't be a subset of a finite collection of vectors.

If W = 0, then clearly W is a particular linear combination of the vectors in set k, so is in span(k).
 

What is the meaning of "span(k) contains more than k" in scientific research?

In scientific research, "span(k) contains more than k" means that the data or results being analyzed have a larger range or variability than expected. This can be an important finding, as it suggests that there may be other factors or variables at play that are influencing the outcome.

Why is it important to understand why span(k) contains more than k?

Understanding why span(k) contains more than k is important because it can provide insight into the underlying factors that may be affecting the data. This can help researchers make more accurate and informed conclusions about their findings and potentially identify new areas for further investigation.

What could be some possible reasons for why span(k) contains more than k?

There are several potential reasons for why span(k) contains more than k. It could be due to natural variation in the data, measurement error, or the presence of additional variables that were not accounted for in the research design. It could also indicate a need for a larger sample size or a more diverse sample.

How can scientists address the issue of span(k) containing more than k?

To address the issue of span(k) containing more than k, scientists can carefully examine their research methods and data to identify any potential sources of variation or error. They can also conduct further experiments or analyses to better understand the underlying factors that may be contributing to the larger range in data.

Is "span(k) contains more than k" always a problem in scientific research?

No, "span(k) contains more than k" is not always a problem in scientific research. In some cases, a larger range in data may be expected or even desirable, as it can provide valuable information about the complexity of the phenomenon being studied. However, it is important for scientists to carefully consider the reasons for the larger range and how it may impact their conclusions.

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