🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

A linear transformation from $V$ to $W$ is a function $T:V\to W$ with the following properties for any $u,v\in V$ and $c\in F$

• $T(u+v)=T\left(u\right)+T\left(v\right)$
• $T\left(cu\right)=cT\left(u\right)$

An inner product is a function $⟨·,·⟩:V\times V\to F$ where $V$ is a vector space over the field $F$, it assigns to each pair $𝐯,𝐰\in V$ a real number $⟨𝐯,𝐰⟩$ such that, for all $𝐮,𝐯,𝐰\in V$ and $\alpha \in 𝐅$ and satisfies:

1. $⟨𝐯,𝐯⟩\ge 0$, with equality if and only if $𝐯=\mathbf{0}$.
2. $⟨𝐯,𝐰⟩=⟨𝐰,𝐯⟩$.
3. $⟨𝐮+𝐯,𝐰⟩=⟨𝐮,𝐰⟩+⟨𝐯,𝐰⟩$,
4. $⟨\alpha 𝐯,𝐰⟩=\alpha ⟨𝐯,𝐰⟩$.

An inner product space is a vector space $V$ over the field $F$ together with an inner product