For a linear transformation $\ltdefn{T}{U}{V}$, the kernel is a subset of the domain $U$. Informally, it is the set of all inputs that the transformation sends to the zero vector of the codomain. It will have some natural connections with the null space of a matrix, so we will keep the same notation, and if you think about your objects, then there should be little confusion. Here is the careful definition.
Suppose $\ltdefn{T}{U}{V}$ is a linear transformation. Then the kernel of $T$ is the set \begin{equation*} \krn{T}=\setparts{\vect{u}\in U}{\lt{T}{\vect{u}}=\zerovector} \end{equation*}
Notice that the kernel of $T$ is just the preimage of $\zerovector$, $\preimage{T}{\zerovector}$ (Definition PI). Here is an example.
We know that the span of a set of vectors is always a subspace (Theorem SSS), so the kernel computed in Example NKAO is also a subspace. This is no accident, the kernel of a linear transformation is always a subspace.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the kernel of $T$, $\krn{T}$, is a subspace of $U$.
Let us compute another kernel, now that we know in advance that it will be a subspace.
Our next theorem says that if a preimage is a nonempty set then we can construct it by picking any one element and adding on elements of the kernel.
Suppose $\ltdefn{T}{U}{V}$ is a linear transformation and $\vect{v}\in V$. If the preimage $\preimage{T}{\vect{v}}$ is nonempty, and $\vect{u}\in\preimage{T}{\vect{v}}$ then \begin{equation*} \preimage{T}{\vect{v}}= \setparts{\vect{u}+\vect{z}}{\vect{z}\in\krn{T}} =\vect{u}+\krn{T} \end{equation*}
This theorem, and its proof, should remind you very much of Theorem PSPHS. Additionally, you might go back and review Example SPIAS. Can you tell now which is the only preimage to be a subspace?
Here is the cartoon which describes the “many-to-one” behavior of a typical linear transformation. Presume that $\lt{T}{\vect{u}_i}=\vect{v}_i$, for $i=1,2,3$, and as guaranteed by Theorem LTTZZ, $\lt{T}{\zerovector_U}=\zerovector_V$. Then four pre-images are depicted, each labeled slightly different. $\preimage{T}{\vect{v}_2}$ is the most general, employing Theorem KPI to provide two equal descriptions of the set. The most unusual is $\preimage{T}{\zerovector_V}$ which is equal to the kernel, $\krn{T}$, and hence is a subspace (by Theorem KLTS). The subdivisions of the domain, $U$, are meant to suggest the partioning of the domain by the collection of pre-images. It also suggests that each pre-image is of similar size or structure, since each is a “shifted” copy of the kernel. Notice that we cannot speak of the dimension of a pre-image, since it is almost never a subspace. Also notice that $\vect{x},\,\vect{y}\in V$ are elements of the codomain with empty pre-images.
The next theorem is one we will cite frequently, as it characterizes injections by the size of the kernel.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then $T$ is injective if and only if the kernel of $T$ is trivial, $\krn{T}=\set{\zerovector}$.
You might begin to think about how Diagram KPI would change if the linear transformation is injective, which would make the kernel trivial by Theorem KILT.