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 \[ \lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert \]