[Math] Is the kernel of a matrix its nullspace and nothing more? And what is the term for the dimension of a kernel

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Are null space and kernel perfect synonyms? If so, why are there two different terms for them?

Also, does the term "nullity" have a "kernel" equivalent? Nullity refers to the dimension of the null space (and thus dimension of the kernel), and seems very important, so is there another term I should be aware of that refers to nullity?

Best Answer

The short answer is that "kernel" and "null space" mean almost exactly the same thing, and you will likely never cause confusion by using the two terms interchangeably.

That being said, there are are some nuances, and I would not say that the two terms are "perfect synonyms". In my experience, "kernel" is a more general term than "null space". The term "kernel" is used to refer to a set in the domain of a homomorphism, where the homomorphism could be of any type; while the term "null space" refers specifically to a set in the domain of a linear transformation between vector spaces (a vector space homomorphism).

This seems to be consistent with a quick check of reliable sources on the internet. For example, if we examine the definitions on MathWorld, we find:

Definition: For any function $f:A\to B$ (where $A$ and $B$ are any sets), the kernel (also called the null space) is defined by $$\ker(f) = \{x : x\in A\text{ such that }f(x) = 0\}.$$

Definition: If $T$ is a linear transformation of $\mathbb{R}^n$, then the null space $\operatorname{Null}(T)$, also called the kernel, is the set of all vectors $\mathbf{X}$ such that $$T(\mathbf{X}) = \mathbf{0}. $$

Per these definitions, the kernel and null space are exactly the same thing, though I would note that the definition of the null space refers only to linear transformations on real vector spaces, while the definition of the kernel applies to maps between any two sets (though, presumably, $B$ must contain some kind of zero element).

Wikipedia gives more detailed definitions. Taking a somewhat general approach from the page kernel (algebra), we have

Definition: Let $A$ and $B$ be algebraic structures of a given type and let $f$ be a homomorphism of that type from $A$ to $B$. Then the kernel of $f$ is the subset of the direct product $A \times A$ consisting of all those ordered pairs of elements of $A$ whose components are both mapped by $f$ to the same element in $B$. The kernel is usually denoted $\ker f$ (or a variation). In symbols: $$\ker f=\{(a,a')\in A\times A:f(a)=f(a')\}. $$

In the cases where elements of $B$ have inverses, we have $$ f(a) = f(a') \implies 1_B = -f(a) + f(a') = f(-a+a'), $$ where $+$ denotes the appropriate operation in $A$ or $B$ (depending on context), $1_B$ is the identity element in $B$ (note that $\mathbf{0}$ is the usual notation for the identity element in a vector space), and the last equality follows from properties of homomorphisms.

Wikipedia also gives the definition of a null space, under the heading kernel (linear algebra):

Definition: in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map $L : V \to W$ between two vector spaces $V$ and $W$, is the set of all elements $v$ of $V$ for which $L(v) = 0$, where 0 denotes the zero vector in $W$. That is, in set-builder notation, $$\ker(L)=\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \}. $$

Per these definitions, the kernel and null space are not quite the same thing. In particular, the term null space applies only to linear transformations between vector spaces, while the term kernel is much more general.

Generally speaking, you will likely not encounter the term "null space" except in the context of linear maps between vector spaces. However, you would never be misunderstood if you called it the "kernel" of that map, instead. On the other hand, if you were to refer to the "null space" of a group homomorphism, you might get a raised eyebrow or two. In that context, you are probably better off using the term "kernel".

Regarding your question about the existence of an equivalent term to "nullity," I don't know the answer, though I suspect that it is "No."

When working with vector spaces, the dimension of those vector spaces are both important and relatively simple to define and determine. The rank-nullity theorem relates the dimensions of the image and the null space, hence it is useful to have a term like "nullity" to abbreviate and simplify communication.

On the other hand, when working with more general algebraic structures, the notion of "dimension" is a lot more subtle, and does not necessarily work as you would like. For example, I am not aware of a general version of the rank-nullity theorem which can be applied to module homomorphisms (though I suspect that some weak generalization might exist?), let alone group homomorphisms. Since the "dimension" of the kernel of an arbitrary homomorphism may not even be defined, I suspect that there is no generally used term for the dimension of the kernel of a map.

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