A vector v is in the kernel of a matrix A if and only if Av=0. Thus, the kernel is the span of all these vectors.
Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0.
For example the kernel of this matrix (call it A)
$ \begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 1\end{bmatrix} $
is the following matrix, where s can be any number:
$ \begin{bmatrix} 0 \\ -s\\ 2s\end{bmatrix} $
Verification using matrix multiplaction: the first entry is $ 0*1-s*0+2s*0=0 $ and the second entry is $ 0*0-s*2+2s*1=0 $.
$ \begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 1\end{bmatrix}* \begin{bmatrix} 0 \\ -s\\ 2s\end{bmatrix}= \begin{bmatrix} 0 \\ 0\end{bmatrix} $
A related concept is that of image of a matrix A.
The dimensions of the image and the kernel of A are related in the Rank Nullity Theorem
