Definition:Kernel of Linear Transformation/Vector Space

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

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

 * Definition:Null Space


 * Kernel of Linear Transformation contains Zero Vector