Linear-2 7

Inner product #definition

\begin{matrix} Let V be a vector space over F (F \in {R, C}) \\ ⟨, ⟩ : V \times V \to F is called inner product if \\ 1. Linearity in the first argument: ⟨ v + α u, w ⟩ = ⟨ v, w ⟩ + α ⟨ u, w ⟩ \\ 2. Conjugate symmetry (Hermitian): ⟨ v, u ⟩ = \overset{―}{⟨ u, v ⟩} \\ 3. Positive-definiteness: \forall v \neq 0 : ⟨ v, v ⟩ > 0 \\ 3.1 v = 0 ⟺ ⟨ v, v ⟩ = 0 \\ V is then called an inner product space \end{matrix}

\begin{matrix} Let V = R^{n} \\ Standard inner product is ⟨ v, u ⟩ = v^{T} u = \sum_{i = 1}^{n} v_{i} u_{i} \\ ⟨ v + α u, w ⟩ = (v + α u)^{T} w = v^{T} w + α u^{T} w = ⟨ v, w ⟩ + α ⟨ u, w ⟩ \\ ⟨ v, w ⟩ = v^{T} w = (v^{T} w)^{T} = w^{T} v = \overset{―}{w^{T} v} = \overset{―}{⟨ w, v ⟩} \\ v \neq 0 ⟹ ⟨ v, v ⟩ = v^{T} v = \sum_{i = 1}^{n} v_{i}^{2} > 0 \\ v = 0 ⟺ \sum_{i = 1}^{n} v_{i}^{2} = 0 ⟺ v^{T} v = 0 ⟺ ⟨ v, v ⟩ = 0 \\ Let V = C^{n} \\ Standard inner product is ⟨ v, u ⟩ = v^{T} \overset{―}{u} = \sum_{i = 1}^{n} v_{i} \overset{―}{u_{i}} \\ Let V \in F^{n \times n} \\ Standard inner product is ⟨ A, B ⟩ = t r (A B^{*}) = t r (A {\overset{―}{B}}^{T}) \end{matrix}

\begin{matrix} \forall v \in V : ⟨ 0_{V}, v ⟩ = ⟨ 0_{F} \cdot 0_{V}, v ⟩ = 0_{F \cdot ⟨ 0_{V}, v ⟩} = 0_{F} \\ \forall v \in V : ⟨ v, 0_{V} ⟩ = \overset{―}{⟨ 0_{V}, v ⟩} = ⟨ 0_{V}, v ⟩ = 0_{F} \\ ⟨ v, w + α u ⟩ = \overset{―}{⟨ w + α u, v ⟩} = \overset{―}{⟨ w, v ⟩} + \overset{―}{α} \cdot \overset{―}{⟨ u, v ⟩} = \overset{―}{\overset{―}{⟨ v, w ⟩}} + \overset{―}{α} \cdot \overset{―}{\overset{―}{⟨ v, u ⟩}} = \\ = ⟨ v, w ⟩ + \overset{―}{α} \cdot ⟨ v, u ⟩ \end{matrix}

\begin{matrix} Let v \in V : \forall u \in V : ⟨ v, u ⟩ = 0 \\ Then v = 0 \\ Proof: \\ \forall u \in V : ⟨ v, u ⟩ = 0 ⟹ ⟨ v, v ⟩ = 0 ⟹ v = 0 \end{matrix}

\begin{matrix} Let V over F \\ ‖ ‖ : V \times V \to F is called a norm if \\ 1. v \neq 0 ⟹ ‖ v ‖ > 0 \\ 2. v = 0 ⟺ ‖ v ‖ = 0 \\ 3. ‖ α v ‖ = | α | \cdot ‖ v ‖ \\ 4. ‖ v + u ‖ \leq ‖ v ‖ + ‖ u ‖ \end{matrix}

\begin{matrix} \forall v \in V : ‖ v ‖ = \sqrt{⟨ v, v ⟩} \\ This norm will be used throughout the course \end{matrix}

\begin{matrix} p : V \times V \to R is called a metric if \\ 1. p (v, u) \geq 0 \\ 2. v = u ⟺ p (v, u) = 0 \\ 3. p (v, u) = p (u, v) \\ 4. p (v, u) \leq (p, w) + p (w, u) \end{matrix}

\begin{matrix} \forall v, u \in V : p (v, u) = ‖ v - u ‖ \end{matrix}

\begin{matrix} v, u are called orthogonal iff ⟨ v, u ⟩ = 0 \end{matrix}

\begin{matrix} Let S \subseteq V \\ S is called orthogonal set iff \forall v, u \in S : ⟨ v, u ⟩ = 0 \end{matrix}

\begin{matrix} v is called normal iff ‖ v ‖ = 1 \end{matrix}

\begin{matrix} Let S \subseteq V be a orthogonal set \\ S is then called orthonormal iff \forall v \in S : ‖ v ‖ = 1 \end{matrix}