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

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

An inner product is a function $$ ⟨\cdot <!--\; \cdot \; -->,\cdot <!--\; \cdot \; -->⟩:V\times <!--\; \times \; -->V\to <!--\; \to \; -->F$$ where $$ V$$ is a vector space over the field $$ F$$, it assigns to each pair $$ \mathbf{v},\mathbf{w}\in <!--\; \in \; -->V$$ a real number $$ ⟨<!--\; ⟨\; -->\mathbf{v},\mathbf{w}⟩<!--\; ⟩\; -->$$ such that, for all $$ \mathbf{u},\mathbf{v},\mathbf{w}\in <!--\; \in \; -->V$$ and $$ \mathrm{\alpha <!--\; \alpha \; -->}\in <!--\; \in \; -->\mathbf{F}$$ and satisfies:

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

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