Subspace, vectorspace, nullspace, columnspace

[V0ODO0CH1LD]

I am wondering how to organise all of those concepts in my head.
should i think of it like:

subspace > vectorspace > nullspace, columnspace

kind of like columnspaces and nullspaces are valid vectorspaces, and all of those are valid subspaces. is a vector space a columnspace? except its definition depends on vectors?

also I realized recently that the nullspace of say Ax=0 could be defined in lower dimensions then the columnspace of matrix A. does that mean that those are completely unrelated concepts, except for coming from the same place?

 

[SteveL27]

I am wondering how to organise all of those concepts in my head.
should i think of it like:

subspace > vectorspace > nullspace, columnspace

kind of like columnspaces and nullspaces are valid vectorspaces, and all of those are valid subspaces. is a vector space a columnspace? except its definition depends on vectors?

also I realized recently that the nullspace of say Ax=0 could be defined in lower dimensions then the columnspace of matrix A. does that mean that those are completely unrelated concepts, except for coming from the same place?

Order of concepts:

Vector space; subspace; linear transformation; kernel (nullspace); image (column space).

The reason vector space is first, is that a subspace is a subset of a vector space that's also a vector space. So you have to understand what a vector space is before you can understand a subspace.

And you have to throw linear transformations in there because the concepts of kernel and image are relative to some linear transformation.

Hope this helps.

 

[algebrat]

Vector space, okay, you know what that is.

Subspace sits inside the vector space. (However, a subspace is a vector space. And a vector space is a subspace of itself. But that's just weird ways to think.)

Two examples of subspaces are null space and column space.

Visually, I keep the null space in the domain, and the column space in the range.

The null space does not have to be smaller than the column space. But the image of the null space is definitely small, wouldn't you agree? The column space could be small, and it is no bigger than the range. (The column space is the image.) Likewise, the null space is no larger than the domain, but the null space is possibly small too.

But theory tells us that the dimension of the null space plus the dimension of the column space add up to the dimension of the domain. (But remember, the column space stays in the range.)

So I guess the column space, which sits in the range, is no bigger than the domain also.

 

[algebrat]

I reread your question, I realized I should be a little more specific in places.

Vector space has a definition, and it's sort of the space which we use to study linear algebra. Subspace is defined as something sitting inside.

Now, if we have a matrix A, then...

the column space is the image of A. The null space is all the elements in the domain which are mapped to 0 by A. It's sometimes called the kernel of A, or the preimage of 0 (under the map A).

Because of the way we defined everything, and that somebody picked out the subject material in just the right way, we can quickly show that the column space and row space are subspaces (of the domain and range respectively).

So vector spaces are not typically column spaces, it's more like, vector spaces are the arenas in which we find these objects. And A is a map between these arenas.

 

[blue_raver22]

Yes, you are correct in your understanding of how these concepts are related. A subspace is a subset of a vector space that also satisfies the properties of a vector space, so all subspaces are also vector spaces. The nullspace and columnspace are both subspaces of a vector space, but they have specific definitions and properties that make them unique. The nullspace is the set of all vectors in the domain that are mapped to the zero vector in the range by a linear transformation, while the columnspace is the span of the columns of a matrix. Both of these are valid vector spaces, but they have different bases and dimensions.

A vector space can be thought of as a generalization of a columnspace, as it is a set of vectors that can be added and multiplied by scalars. However, the definition of a vector space is not limited to columns of a matrix, it can include any set of vectors that satisfy the properties of a vector space. Therefore, a vector space is not necessarily a columnspace, but a columnspace is always a vector space.

The fact that the nullspace and columnspace may have different dimensions does not mean they are unrelated. In fact, they are closely related as they are both subspaces of the same vector space. The difference in dimensions is due to the different definitions and properties of each subspace.

In summary, it is helpful to think of these concepts in a hierarchical way, with vector space being the broadest concept and subspaces being more specific types of vector spaces. The nullspace and columnspace are both valid subspaces, but they have unique definitions and properties that make them distinct from each other.