Linear mapping/Image and preimage/Linear subspaces/Fact/Proof/Exercise

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

$\varphi \colon V\longrightarrow W$

be a linear map. Prove the following facts.

1. For a linear subspace ${}S\subseteq V$, also the image ${}\varphi (S)$ is a linear subspace of ${}W$.
2. In particular, the image

   ${}\operatorname {Im} \varphi =\varphi (V)\,$

   of the map is a subspace of ${}W$.

3. For a linear subspace ${}T\subseteq W$, also the preimage ${}\varphi ^{-1}(T)$ is a linear subspace of ${}V$.
4. In particular, ${}\varphi ^{-1}(0)$ is a subspace of ${}V$.

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