# What is a "kernel" in linear algebra?

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