How do you tell what a vector space will look like from it's spanning set.

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SUMMARY

This discussion focuses on visualizing vector spaces through their spanning sets. Participants emphasize that a minimal spanning set consists of linearly independent vectors, which can be represented as arrows in a graph. The presence of an inner product space allows for the determination of vector length and orientation, enabling accurate plotting in Euclidean space. Understanding these concepts is crucial for determining the dimensionality and visual characteristics of vector spaces.

PREREQUISITES
  • Understanding of vector spaces and spanning sets
  • Knowledge of linear independence and dependence
  • Familiarity with inner product spaces
  • Basic graphing skills in Euclidean space
NEXT STEPS
  • Explore the properties of linear independence in vector spaces
  • Learn how to visualize vector fields using software tools
  • Study the implications of inner product spaces on vector representation
  • Investigate dimensionality and its relation to spanning sets
USEFUL FOR

Students and educators in mathematics, particularly those studying linear algebra, as well as professionals in fields requiring geometric interpretations of vector spaces.

brandy
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I'm having a hard time visualising them.
How do you plot vector fields
How do you plot spanning sets?
How do you tell if something spans a plane, 3d, a line or more dimensions.
 
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Given a spanning set, determine how many of them are dependent on the others and so can be dropped from the spanning set. The number of vectors in a "minimal" spanning set (so all vectors in the set are independent) is the dimension of the space they span.
 
I know that, and it's not really my question unless I misunderstand the applications of what you just said.
My question pertains to the visible nature of it.
 
brandy said:
I know that, and it's not really my question unless I misunderstand the applications of what you just said.
My question pertains to the visible nature of it.

Hey brandy.

In your vector space can visualize the minimal spanning set as a bunch of arrows that represent the relevant information for the vector space depending on what structure is represented and how they relate to the components in the 'vector'.

If you have an inner product space included in your vector space, you can find the length and orientation of your vectors which means you can actually plot these vectors on a graph of some sort (Euclidean in whatever dimension) and this can be done no matter what kind of vector space you have (as long as it has a valid inner product).

So in the above case, you will get n arrows that are linearly independent and these represent the visual characteristics of the minimal spanning set.
 

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