Linear Transformation
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 $$c \in F$$
• $$T \left( u + v \right) = T \left( u \right) + T \left( v \right)$$
• $$T \left( c u \right) = c T \left( u \right)$$
Inner Product
An inner product is a function $$\left\langle \cdot , \cdot \right\rangle : V \times V \to F$$ where $$V$$ is a vector space over the field $$F$$, it assigns to each pair $$\mathbf{v}, \mathbf{w} \in V$$ a real number $$\langle\mathbf{v}, \mathbf{w}\rangle$$ such that, for all $$\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$$ and $$\alpha \in \mathbf{F}$$ and satisfies:
1. $$\langle\mathbf{v}, \mathbf{v}\rangle \geq 0$$, with equality if and only if $$\mathbf{v}=\mathbf{0}$$.
2. $$\langle\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{w}, \mathbf{v}\rangle$$.
3. $$\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle$$,
4. $$\langle\alpha \mathbf{v}, \mathbf{w}\rangle=\alpha\langle\mathbf{v}, \mathbf{w}\rangle$$.
Inner Product Space
An inner product space is a vector space $$V$$ over the field $$F$$ toegether with an inner product
The Dot Product in Rn Forms an Inner Product
The dot product in $$\mathbb{ R } ^ n$$ forms an inner product
Schwarz Inequality
For all $$x, y \in \mathbb{ R } ^ n$$ we have $\left\lvert \left\langle x, y \right\rangle \right\rvert \le \lVert x \rVert \lVert y \rVert$ and $$\left\lvert \left\langle x, y \right\rangle \right\rvert = \lVert x \rVert \lVert y \rVert$$ if and only if $$x, y$$ are collinear
The Norm Satisfies the Triangle Inequality
For any $$x, y \in \mathbb{ R } ^ n$$ we have $\lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert$