Definition:Kernel of Linear Transformation/Vector Space

From ProofWiki
Jump to navigation Jump to search

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


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 known as

The kernel of a linear transformation $T$ on a vector space is also known as the null space on $T$.

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.