From: hackbard Date: Thu, 9 Feb 2012 20:49:56 +0000 (+0100) Subject: fixed outer product, redo inner product! X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=dcfe9181a03b1eff2ec41d207241647e7392f2fd;p=lectures%2Flatex.git fixed outer product, redo inner product! --- diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex index 9ff2b4b..79e4ec9 100644 --- a/physics_compact/math_app.tex +++ b/physics_compact/math_app.tex @@ -74,14 +74,14 @@ In doing so, conjugacy is associated with duality. \end{remark} \begin{definition} -If $\vec{u}\in U$, $\vec{v}\in V$ and $\vec{v}^{\dagger}\in V^{\dagger}$ are vectors within the respective vector spaces and $V^{\dagger}$ is the dual space of $V$, -the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{v}^{\dagger}$ and $\vec{u}$, +If $\vec{u}\in U$, $\vec{v}\in V$ are vectors within the respective vector spaces and $\vec{y}^{\dagger}\in V^{\dagger}$ is a linear functional of the dual space $V^{\dagger}$ of $V$, +the outer product $\vec{u}\otimes\vec{v}$ is defined as the tensor product of $\vec{y}^{\dagger}$ and $\vec{u}$, which constitutes a map $A:V\rightarrow U$ by \begin{equation} -\vec{v}\mapsto\vec{v}^{\dagger}(\vec{v})\vec{u} +\vec{v}\mapsto\vec{y}^{\dagger}(\vec{v})\vec{u} \text{ ,} \end{equation} -where $\vec{v}^{\dagger}(\vec{v})$ denotes the linear functional $\vec{v}^{\dagger}\in V^{\dagger}$ on $V$ when evaluated at $\vec{v}\in V$, a scalar that in turn is multiplied with $\vec{u}\in U$. +where $\vec{y}^{\dagger}(\vec{v})$ denotes the linear functional $\vec{y}^{\dagger}\in V^{\dagger}$ on $V$ when evaluated at $\vec{v}\in V$, a scalar that in turn is multiplied with $\vec{u}\in U$. In matrix formalism, with respect to a given basis ${\vec{e}_i}$ of $\vec{u}$ and ${\vec{e}'_i}$ of $\vec{v}$, if $\vec{u}=\sum_i^m \vec{e}_iu_i$ and $\vec{v}=\sum_i^n\vec{e}'_iv_i$,