Differnece between the Kernel and the Nullspace?

[Xyius]

What is the distinct difference between the kernel and the null space? They both have the same definition, namely, Ax=0.

 

[Fredrik]

There's no difference. Not in linear algebra (or functional analysis) anyway, but the word "kernel" is used for different things in other areas of mathematics, and "null space" isn't (as far as I know).

 

[Xyius]

There's no difference. Not in linear algebra (or functional analysis) anyway, but the word "kernel" is used for different things in other areas of mathematics, and "null space" isn't (as far as I know).

Thanks a lot! Exactly what I wanted to hear :]

 

[Newtime]

There's no difference. Not in linear algebra (or functional analysis) anyway, but the word "kernel" is used for different things in other areas of mathematics, and "null space" isn't (as far as I know).

Although the concepts are intimately related, it is good to keep in mind when you are referring to a transformation and when you are referring to it's matrix representation. For instance, if your linear transformation is the partial derivative with respect to x of some function of x and y up to order n, then the kernel is all polynomials (of order n) with no x term but the null space of the representation of this transformation is some subspace of R^n. Of course, these vectors in the subspace of R^n are exactly understood as polynomials in 2 variables of degree at most n, since the matrix is the representation of the transformation, but like I said, in this case, the kernel and the nullspace are different since one consists of polynomials and the other vectors in R^n, though there is a bijection between them since one is an exact representation of the other.

If my reasoning is flawed, please correct me, but this is generally what I've come across.

 

[Fredrik]

So you're saying that "kernel" is used for linear operators, but not for the corresponding matrices, and that "null space" is used for matrices, but not for the corresponding linear operators? I don't think that terminology is standard.

 

[Newtime]

So you're saying that "kernel" is used for linear operators, but not for the corresponding matrices, and that "null space" is used for matrices, but not for the corresponding linear operators? I don't think that terminology is standard.

More or less, yes. Perhaps it is not standard, but I've always seen it done this way and in any case I suppose the argument is largely semantical since the ideas are identical.

 

[Fredrik]

OK, that terminology might be common, I don't know. Axler isn't using it. Page 41 defines the null space of a linear operator.

 

[Newtime]

OK, that terminology might be common, I don't know. Axler isn't using it. Page 41 defines the null space of a linear operator.

Yes but isn't the notation "L(V, W)" referring to the set of all linear operators from the vector space V to the vector space W? In general, linear operators need not take vector spaces to vector spaces, but in the context of this book, it makes complete sense. Again though, we're describing the same idea, so the word itself doesn't matter as long as the context makes it clear. (I hope I don't seem argumentative, I too want to understand where my reasoning is wrong if indeed it is.)

 

[Fredrik]

Yes but isn't the notation "L(V, W)" referring to the set of all linear operators from the vector space V to the vector space W?

Yes, that's precisely my point. You said that when $T\in\mathcal L(V,W)$, the set $\{x\in V|Tx=0\}$ is called the "kernel" of T, and not the "null space" of T. So this example shows that there's at least one author that doesn't use the same terminology as you. (There may be others who agree with you. My point is just that your terminology certainly isn't used by everyone).

In general, linear operators need not take vector spaces to vector spaces,

I haven't seen the term "linear operator" used in any other context, but I suppose we could use it for functions between modules as well. (A module is essentially a vector space over a ring, instead of over a field). I don't see how the term could make sense in any other context.

 

[Newtime]

Yes, that's precisely my point. You said that when $T\in\mathcal L(V,W)$, the set $\{x\in V|Tx=0\}$ is called the "kernel" of T, and not the "null space" of T. So this example shows that there's at least one author that doesn't use the same terminology as you. (There may be others who agree with you. My point is just that your terminology certainly isn't used by everyone).


I haven't seen the term "linear operator" used in any other context, but I suppose we could use it for functions between modules as well. (A module is essentially a vector space over a ring, instead of over a field). I don't see how the term could make sense in any other context.

Right. I was also thinking about homomorphisms between additive groups.

 

[pbandjay]

My professors have always used only the term Kernel, in both linear algebra and abstract algebra. Of course when referring to the vector space R^n and matrices, I can see how the term Null Space would go well with Vector Space, Column Space, etc.

 

[Landau]

Null space and kernel are synonyms, but indeed null space is usually only used in linear algebra, whereas kernel is also used in other (similar) contexts.
Further, I don't want to go offtopic, but:

(A module is essentially a vector space over a ring, instead of over a field). I don't see how the term could make sense in any other context.

There are (at least) two generalizations of the notion of "kernel":
* kernel in category theory.
* kernel in universal algebra.

 

[Fredrik]

I was actually talking about the term "linear operator", but thanks anyway. I'm actually interested in all this "abstract nonsense" (and I don't know much about it).

 

[Landau]

Ah, my bad, reading your post again I understand what you meant :)

 

[shelovesmath]

I'm bumping this question. I'm wondering if there is a difference as far as the kernel being a set and the null space being a subspace. Is the kernel actually a subspace itself of the vector space it is mapping from? Or is it only just a set of vectors that maps to 0 on another vector space?

 

[shelovesmath]

I'm bumping this question. I'm wondering if there is a difference as far as the kernel being a set and the null space being a subspace. Is the kernel actually a subspace itself of the vector space it is mapping from? Or is it only just a set of vectors that maps to 0 on another vector space?

Nevermind, stupid question.

 

[Deveno]

a null space is a kernel. not all kernels are null spaces.

if you are talking about within the context of linear algebra, they are equivalent concepts.

but "kernels" can be constructed for different kinds of structure-preserving maps than linear transformations.

 

[HallsofIvy]

I'm bumping this question. I'm wondering if there is a difference as far as the kernel being a set and the null space being a subspace. Is the kernel actually a subspace itself of the vector space it is mapping from? Or is it only just a set of vectors that maps to 0 on another vector space?

As others have said, in Linear Algebra, "kernel" (of a linear transformation) and "nullspace" are the same thing. One can show that the kernel of any linear transformation, from one vector space to another, is subspace (perhaps trivial) of the domain space.

If u and v are in the nullspace of L, then L(u+ v)= L(u)+ L(v)= 0+ 0= 0 so the null space is closed under addition. If u is in the nullspace of L and k is any scalar, then L(ku)= kL(u)= 0 so the null space is also closed under scalar multiplication. Therefore it is a subspace and the name "null space" is justified. I suppose one could use the term "kernel" for sets, not subspaces, such that f(u)= 0 for some function f that is NOT a linear transformation, but the "linearity" is, after all, the whole point of vector spaces.

Given a function from one algebraic object to another, other than vector spaces, the term "kernel" is used to mean the set of points in one that are mapped into the additive identity of the other.