Definition:Kernel of Linear Transformation/Vector Space

Definition
Let $\left({\mathbf V,+,\times}\right)$ be a vector space.

Likewise let $\left({\mathbf V\,',+,\times}\right)$ 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:


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

Also see

 * Null Space
 * Kernel of Linear Transformation Contains Zero Vector