Subspace vs Subset: Understanding the Relationship

[Max.Planck]

Hi,

A quick question:
Does a set need to be a subset to be a subspace of some vector space?

 

[micromass]

Hi,

A quick question:
Does a set need to be a subset to be a subspace of some vector space?

Yes, a subspace of a vector space V necessarily needs to be a subset.

 

[HallsofIvy]

A vector space is a set of objects with "addition" and "scalar multiplication" defined. A "subspace" is a subset with the "inherited" addition and scalar multiplication.

 

[Max.Planck]

A vector space is a set of objects with "addition" and "scalar multiplication" defined. A "subspace" is a subset with the "inherited" addition and scalar multiplication.

Thank you!

 

[blue_raver22]

The relationship between subspace and subset can be a bit confusing, but it is an important concept to understand in mathematics and science. To answer your question, a set does not necessarily need to be a subset to be a subspace of a vector space.

A subset is simply a collection of elements that are contained within a larger set. For example, the set {1,2,3} is a subset of the set of natural numbers {1,2,3,4,5...}. In this case, all of the elements in the subset are also elements of the larger set.

On the other hand, a subspace is a subset of a vector space that also satisfies certain properties. These properties include being closed under scalar multiplication and vector addition. This means that if you take any element from the subspace and multiply it by a scalar or add it to another element in the subspace, the result will still be within the subspace.

So, while a subspace is technically a subset, it is a special type of subset that has additional properties. This means that not all subsets are subspaces, but all subspaces are subsets.

I hope this helps clarify the relationship between subspace and subset. It is an important distinction to understand in order to effectively work with vector spaces and their subspaces in scientific research and applications.