Linear mapping/Image and preimage/Linear subspaces/Fact

Let ${}K$ be a field, let ${}V$ and ${}W$ denote ${}K$ -vector spaces, and let

\varphi \colon V\longrightarrow W

be a ${}K$ -linear mapping. Then the following hold.

For a linear subspace ${}S\subseteq V$ , the image ${}\varphi (S)$ is a linear subspace of ${}W$ .
In particular, the image ${}\operatorname {Im} \varphi =\varphi (V)$ of the mapping is a linear subspace of ${}W$ .
For a linear subspace ${}T\subseteq W$ , the preimage ${}\varphi ^{-1}(T)$ is a linear subspace of ${}V$ .
In particular, ${}\varphi ^{-1}(0)$ is a linear subspace of ${}V$ .