Endomorphism/Simple properties of invariant subspaces/Exercise

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$. Show the following properties.

1. The zero space ${}0\subseteq V$ is ${}\varphi$-invariant.
2. ${}V$ is ${}\varphi$-invariant.
3. Eigenspaces are ${}\varphi$-invariant.
4. Let ${}U_{1},U_{2}\subseteq V$ be ${}\varphi$-invariant linear subspaces. Then also ${}U_{1}\cap U_{2}$ and ${}U_{1}+U_{2}$ are ${}\varphi$-invariant.
5. Let ${}U\subseteq V$ be a ${}\varphi$-invariant linear subspace. Then also the image space ${}\varphi (U)$ and the preimage space ${}\varphi ^{-1}(U)$ are ${}\varphi$-invariant.

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