Definition of Kernel and Image

Let $f : V \to W$ be a linear mapping. Then the image of $f$ is defined as

\[

\text{Im}(f) = f(V) = \{f(x) \mid x \in V\},

\]

and the kernel of $f$ is defined as

\[

\text{Ker}(f) = \{x \in V \mid f(x) = 0\}.

\]