Definition:Kernel of Linear Transformation

Definition

Let $\phi: G \to H$ be a linear transformation where $G$ and $H$ are $R$-modules.

Let $e_H$ be the identity of $H$.

The kernel of $\phi$ defined as:

$\ker\left({\phi}\right) := \phi^{-1} \left({\left\{{e_H}\right\}}\right)$.

where $\phi^{-1}\left({S}\right)$ denotes the preimage of $S$ under $\phi$.

In Vector Spaces

Let $\mathbf V$ be $\R^n$, or any other vector space.

Likewise let $\mathbf V\,'$ be $\R^m$, or any other vector space, with a zero $\mathbf 0\,'$.

Let $T: \mathbf V \to \mathbf V\,'$ be a linear transformation.

Then the kernel of $T$ is defined as:

$\ker \left({T}\right) := T^{-1} \left({\left\{\mathbf 0'\right\}}\right) = \left \{ {\mathbf x \in \mathbf V: T\left({\mathbf x}\right) = \mathbf 0'} \right \}$

Also see

• Null Space
• Kernel of Linear Transformation Contains Zero Vector

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$