Definition:Kernel of Linear Transformation

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This page is about Kernel in the context of Linear Algebra. For other uses, see Kernel.


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 $\phi$ is defined as:

$\map \ker \phi := \phi^{-1} \sqbrk {\set {e_H} }$

where $\phi^{-1} \sqbrk S$ denotes the preimage of $S$ under $\phi$.

In Vector Space

Let $\struct {\mathbf V, +, \times}$ be a vector space.

Let $\struct {\mathbf V', +, \times}$ be a vector space whose zero vector is $\mathbf 0'$.

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

Then the kernel of $T$ is defined as:

$\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$

Also denoted as

The notation $\map {\mathrm {Ker} } \phi$ can sometimes be seen for the kernel of $\phi$.

It can also be presented as $\ker \phi$ or $\operatorname {Ker} \phi$, that is, without the parenthesis indicating a mapping.

Also see

  • Results about kernels of linear transformations can be found here.