Isometry/Same space/Characterizations with orthonormal basis/Exercise

Let $V$ be a Euclidean vector space, and let

$\phi :V⟶V$

denote a linear mapping. Show that the following statements are equivalent.

1. $\phi$ is an isometry.
2. For every vector $v$ with $\Vert v\Vert =1$, we also have $\Vert \phi (v)\Vert =1$.
3. For every orthonormal basis $u_i,i=1,\dots ,n$, also $\phi (u_i),i=1,\dots ,n$, is an orthonormal basis.
4. There exists an orthonormal basis $u_i,i=1,\dots ,n$, such that also $\phi (u_i),i=1,\dots ,n$, is an orthonormal basis.

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