| Version 7 |
Version 6 |
| Let $T\colon V\to W$ be a linear transformation. |
Let $T\colon V\to W$ be a linear transformation. |
| The set of all vectors in $V$ that map to $0$ |
The set of all vectors in $V$ that map to $0$ |
| is called the {\em kernel} (or {\em nullspace}) of $T$, |
is called the {\em kernel} (or {\em nullspace}) of $T$, |
| and is denoted $\ker T$. |
and is denoted $\ker T$. |
| $$ \ker T = |
$$ \ker T = |
| \{\, v \in V\mid T(v)=0\,\} .$$ |
\{\, v \in V\mid T(v)=0\,\} .$$ |
| The kernel is a vector subspace of $V$. |
The kernel is a vector subspace of $V$. |
| Its dimension is called the nullity of $T$. |
Its dimension is called the nullity of $T$. |
| Note that $T$ is injective if and only if $\ker T=\{0\}$. |
|
| When the transformations are given by means of matrices, |
When the transformations are given by means of matrices, |
| the kernel of the matrix $A$ is |
the kernel of the matrix $A$ is |
| $$\ker A=\{\,x\in V \mid Ax=0\,\}.$$ |
$$\ker A=\{\,x\in V \mid Ax=0\,\}.$$ |
