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

[brandy]

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.

 

[HallsofIvy]

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.

 

[brandy]

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.

 

[chiro]

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.

 

[blue_raver22]

I can understand that visualizing vector spaces and their spanning sets can be challenging. However, there are a few ways to approach this problem.

Firstly, it is important to understand that a vector space is a mathematical concept that represents a collection of vectors that can be added and multiplied by scalars. The spanning set of a vector space is a set of vectors that, when combined using scalar multiplication and addition, can generate all the vectors within that space.

To visualize a vector space, one can start by considering the properties of the spanning set. For example, if the spanning set contains only two linearly independent vectors, the vector space will be a plane. If the spanning set contains three linearly independent vectors, the vector space will be three-dimensional.

To plot vector fields, one can use tools such as vector field plots or vector calculators. These tools allow for the visualization of vector fields by plotting arrows representing the magnitude and direction of the vectors at different points in space.

To plot spanning sets, one can use a similar approach by plotting the individual vectors in the set and then connecting them to visualize the span of the set. This can be done in two or three dimensions, depending on the number of vectors in the set.

Finally, to determine the dimensionality of a vector space, one can use the concept of linear independence. If a set of vectors is linearly independent, it will span a space with the same dimensionality as the number of vectors in the set. For example, if a set of three vectors is linearly independent, it will span a three-dimensional space.

In conclusion, visualizing vector spaces and their spanning sets can be challenging, but by understanding their properties and using appropriate tools, it is possible to gain a better understanding of their structure and dimensionality.