Dimostrazione.
Poniamo $\, \vert\vert\mathbf{e}_{1}\vert\vert=1, \, \, <\mathbf{e}_{1}>=\mathbf{W}$, allora
$\mathbf{w}=(\mathbf{v} \cdot \mathbf{e}_{1})\mathbf{e}_{1}$
$\mathbf{w}^{\perp}= \mathbf{v}-\mathbf{w}= \mathbf{v}-(\mathbf{v} \cdot \mathbf{e}_{1})\mathbf{e}_{1}$
1.
$\mathbf{W} \cap \mathbf{W}^{\perp}=<0>$
Se $\exists \, \mathbf{u} \neq 0, \, \mathbf{u} \in \mathbf{W} \cap \mathbf{W}^{\perp} \Rightarrow \, \mathbf{u} \cdot \mathbf{u} =0$ che è impossibile
quindi $\mathbf{W}+\mathbf{W}^{\perp}$ è somma diretta.
2.
Proviamo che $\mathbf{E}=\mathbf{W} \oplus \, \mathbf{W}^{\perp}$.
$\mathbf{w}=(\mathbf{v} \cdot \mathbf{e}_{1})\mathbf{e}_{1}+\cdots+(\mathbf{v} \cdot \mathbf{e}_{t})\mathbf{e}_{t} \in \mathbf{W}$
$\mathbf{w}'=\mathbf{v}-\mathbf{w}=(\mathbf{v}-(\mathbf{v} \cdot \mathbf{e}_{1})...
...-\cdots-(\mathbf{v} \cdot \mathbf{e}_{t})\mathbf{e}_{t}) \in \mathbf{W}^{\perp}$
infatti $\forall i=1,\ldots,t$ si ha
$(\mathbf{v}-(\mathbf{v} \cdot \mathbf{e}_{1})\mathbf{e}_{1}-\cdots-(\mathbf{v} \cdot \mathbf{e}_{t})\mathbf{e}_{t})\mathbf{e}_{i}=0 \quad $
$\vert\vert\mathbf{v}\vert\vert^{2}=\mathbf{v} \cdot \mathbf{v} = (\mathbf{w}+\m...
...}} \cdot \underbrace{\mathbf{w}'}_{\in \mathbf{W}^{\perp}\, \Rightarrow \, =0})$
allora $\quad \vert\vert\mathbf{v}\vert\vert^{2} =\vert\vert\mathbf{w}\vert\vert^{2}+\vert\vert\mathbf{w}'\vert\vert^{2}$
c.v.d.