Let \(T:V\rightarrow W\) be a linear transformation where
\(V\) and \(W\) be vector spaces with scalars coming from the
same field \(\mathbb{F}\).
The kernel of \(T\), denoted by \(\ker(T)\),
is the set of vectors from \(V\)
that gets mapped to the zero vector in \(W\);
that is, \[\ker(T) = \{ v \in V : T(v) = 0_W\}.\]

Examples

Let \(T\) be given by \(T(x) = Ax\) for some \(A \in \mathbb{R}^{m \times n}\).
By definition, the kernel of \(T\) is given by the set of \(x\) such that
\(T(x) = 0\). But \(T(x) = 0\) precisely when \(Ax = 0\).
Therefore, \(\ker(T) = N(A)\), the nullspace of \(A\).

Let \(T\) be a linear transformation from \(P_2\)
to \(\mathbb{R}^2\) given by
\(T(ax^2 + bx + c) = \begin{bmatrix} a+3c \\ a-c\end{bmatrix}\).
The kernel of \(T\) is the set of polynomials \(ax^2+bx+c\) such
that \(\begin{bmatrix} a+3c \\a - c\end{bmatrix}=
\begin{bmatrix} 0\\0\end{bmatrix}\). Solving for
\(a\) and \(c\), we get \(a = c = 0\).
So \(\ker(T) = \{ bx : b \in \mathbb{R}\}\).

Application

Let \(T:V\rightarrow W\) be a linear transformation.
Suppose that you are asked to find all solutions
to \(T(x) = b\) for some \(b \in W\).
If you have found one solution, say \(\tilde{x}\),
then the set of all solutions is
given by \(\{\tilde{x} + \phi : \phi \in \ker(T)\}\).
In other words, knowing a single solution and a description of
the kernel of \(T\) tells you all the solutions to \(T(x) = b\).

When \(T(x)\) is given by \(Ax\), this amounts to saying that
to specify all solutions to \(Ax = b\), we just need one particular solution
and a description of the nullspace of \(A\).