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.

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

Write a proof