17:15 

New Jack
Discover. Acquire. Master. EVOLVE. Repeat
Вопрос: если я получаю сказочное удовольствие от написание этой мути и считаю это чем-то вроде искусства... стоит ли обратится к доктору или проще сразу повеситься?





\documentclass[12pt,a4paper]{article}
\usepackage{amsmath,amssymb}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{url}
\usepackage{float}
\title{Inner product space}
\author{}
\begin{document}
\maketitle
\begin{abstract}
In this article we will get an idea about dot product in vector spaces, specifically in $\mathbb{F}=\mathbb{R}^3$ dimension. Based on this, we will look at the concept of orthogonality (perpendicularity) of two vectors, the way to obtain norm(length) of a vector and the concept of inner product of vectors. Furtherly, we will look at the concept of orthogonal and orthonormal subsets of given vector spaces, finishing with the Gram-Schmidt process $–$ method to construct an orthonormal set of vectors from any given basis which belongs to $V$.
\end{abstract}
\section{General Theory}
\subsection{Initial Definitions}
\cite[Page 45]{1} First of all, lets define a dot product for two vectors $\overrightarrow{v}=\{v_1,v_2,v_3\}$ and $\overrightarrow{u}=\{u_1,u_2,u_3\}$ \qquad $\forall \overrightarrow{v}$, $\overrightarrow{u} \in \mathbb{R}$
$$\overrightarrow{v}\cdot\overrightarrow{u}={v_1}{u_1}+{v_2}{u_2}+{v_3}{v_3}$$
From this we can define a norm$\|\overrightarrow{v}\|$ of a vector $\overrightarrow{v}$
$$\|\overrightarrow{v}\|=\sqrt{\overrightarrow{v}\cdot\overrightarrow{v}}=\sqrt{v^2_1+v^2_2+v^2_3}$$
(Note, that both norm and dot product become constants)/nextline Using these two identities we can obtain the corner between two vectors, by formula:
$$\cos\theta=\frac{\overrightarrow{v}\cdot\overrightarrow{u}}{\|\overrightarrow{v}\|\|\overrightarrow{u}\|}$$
Thus vectors $\overrightarrow{v}$ and $\overrightarrow{u}$ are orthogonal(perpendicular) if and only if $\overrightarrow{v}\cdot\overrightarrow{u}=0$
\subsection{Real inner product}
\cite[Chapter 7.2, page236]{2} \cite{4}
The real inner product of two given vectors $\overrightarrow{v},\overrightarrow{u}\in V$ is a map $$\langle\overrightarrow{v},\overrightarrow{u}\rangle:V\times V\mapsto\mathbb{F}$$
which must satisfy the following axioms:
\begin{itemize}
\item $[a_1]$\textbf{(Symmetric Property)}: \qquad $$\langle\overrightarrow{v},\overrightarrow{u}\rangle=\langle\overrightarrow{u},\overrightarrow{v}\rangle\qquad\forall\overrightarrow{v},\overrightarrow{u}\in\mathbb{V}$$
\item $[a_2]$\textbf{(Linear Property)}: \qquad
$$\langle a\overrightarrow{v},\overrightarrow{u}\rangle = a\langle\overrightarrow{v},\overrightarrow{u}\rangle \qquad \forall \overrightarrow{v},\overrightarrow{u}\in V \qquad \forall a \in \mathbb{F}$$
$$\langle\overrightarrow{v}\oplus\overrightarrow{u},\overrightarrow{w}\rangle = \langle\overrightarrow{v},\overrightarrow{w}\rangle \oplus \langle\overrightarrow{u},\overrightarrow{w}\rangle \qquad \forall\overrightarrow{v},\overrightarrow{u},\overrightarrow{w}\in V$$
\item $[a_3]$\textbf{(Positive Definite Property)} : \qquad $$\langle\overrightarrow{v},\overrightarrow{v}\rangle \geq 0; and \langle\overrightarrow{v},\overrightarrow{v}\rangle = 0 \text{ if and only if } \overrightarrow{v}=\overrightarrow{o}\qquad \forall \overrightarrow{v},\overrightarrow{o} \in V$$
\end{itemize}
\cite[Page 46]{1} A vector space V with real inner product is called a real inner product space and a norm for every vector $\overrightarrow{v}$ in it will be :
$$\|\overrightarrow{v}\|=\sqrt{\langle\overrightarrow{v},\overrightarrow{v}\rangle}$$
\newline
In order to get a real inner product of various cases we use the following formulas\cite[Page46-48]{1}:
\begin{enumerate}
\item When we have real inner product space V=$\mathbb{R}^3$ with usual $\oplus$ and $\cdot$, then for given vectors $\overrightarrow{v}=\{v_1,v_2,v_3,...,v_n\}$ and $\overrightarrow{u}=\{u_1,u_2,u_3,...,u_n\}$, which belong to V, we simply use dot product of two vector in order to obtain their inner product:
$$\langle\overrightarrow{v},\overrightarrow{u}\rangle=v_1 u_1 + v_2 u_2 + v_3 u_3 +...+v_n u_n$$
\item For V=M(n,m) with matrixes A,B $\in$ M(n,m) the inner product is :
$$\langle \overrightarrow{A},\overrightarrow{B} \rangle = Tr(B^T A)$$
with $B^T$ is a transpose of matrix B. Do note that matrix $B^T A$ is a square $n\times n$ matrix, what makes the trace of it obtainable.
\item For given \emph{function space} C[a,b] and \emph{polynomial space} P(x), where $t \in$[a,b] and functions $f(x),g(x) \in C[a,b]$ inner product in C[a,b] is
$$\langle \overrightarrow{f},\overrightarrow{g} \rangle = \int_a^b f(x)g(x)dx$$
\end{enumerate}
\subsection{Orthogonal and Orthonormal Sets}
\cite[Page49]{1}\newline
\textbf{Note}: For given inner product space $V$ and $S$ - set of vectors in it, we define:\newline $S$ to be \emph{orthogonal} set if $\langle\overrightarrow{v},\overrightarrow{u}\rangle=0 \qquad \forall \overrightarrow{v},\overrightarrow{u}\in V$ \newline $S$ to be \emph{orthonormal} is it is an orthogonal set and $\|\overrightarrow{v}\|=1 \qquad \forall \overrightarrow{v}\in V$
\section{Gram-Schmidt process}
\subsection{Theory}
\cite[Page 52-53]{1}
The Gram-Schmidt process is a method of constructing an orthonormal basis for an inner product space from any given basis $S=(\overrightarrow{u_1},\overrightarrow{u_2},\overrightarrow{u_3},...,\overrightarrow{u_n})$ which belongs to it.
\newline
This is a multiple step method and the formula for each Step m is\newline
\newline
$\overrightarrow{w_m}=\overrightarrow{u_m}-proj_{W_m-1}(\overrightarrow{u_m})=\overrightarrow{u_m}-\langle\overrightarrow{u_m},\overrightarrow{v_1}\rangle\overrightarrow{v_1}-\langle\overrightarrow{u_m},\overrightarrow{v_2}\rangle\overrightarrow{v_2}-...-\langle\overrightarrow{u_m},\overrightarrow{w_{m-1}}\rangle\overrightarrow{w_{m-1}}$\newline
$\overrightarrow{v_m}=\frac{1}{\|\overrightarrow{w_k}\|}\overrightarrow{w_k}$\newline
\newline
\begin{figure}[H]
\centering
\includegraphics[scale=1]{Gram–Schmidt_process}\\
\caption{\cite{5}, First two steps of the Gram-Schmidt process}\label{Photo}
\end{figure}

$W_{m-1}$ is the set of all previously obtained vectors $\overrightarrow{v_i}$ and the final $W_n$ is an orthonormal set we were constructing.
\subsection{Example}
\cite[ex 2.10 b), page 257]{3}
Perform the Gram-Schmidt process on the following basis to turn it into orthonormal basis:
S=$\big((1,-1,0),(0,1,0),(2,3,1))$\newline \newline Lets name each vector of the given basis as $\overrightarrow{u_1}=(1,-1,0), \overrightarrow{u_2}=(0,1,0), \overrightarrow{u_3}=(2,3,1)$. Then:
\newline
\newline
\qquad \textbf{Step1} \newline
$\overrightarrow{v_1}=\frac{1}{\|\overrightarrow{u_1}\|}\overrightarrow{u_1}=\frac{1}{\sqrt{1^2+(-1)^2+0}}(1,-1,0)=\frac{1}{\sqrt{2}}(1,-1,0)=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)$
\newline Defining $W_1=span((\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0))$
\newline
\newline
\qquad \textbf{Step2}\newline
$\overrightarrow{w_2}=\overrightarrow{u_2}-proj_{W_1}(\overrightarrow{u_2})=\overrightarrow{u_2}-\langle\overrightarrow{u_2},\overrightarrow{v_1}\rangle\overrightarrow{v_1}=(0,1,0)-(0\cdot\frac{1}{\sqrt{2}}+1\cdot(-\frac{1}{\sqrt{2}})+0\cdot\frac{1}{\sqrt{2}})\cdot\frac{1}{\sqrt{2}}\cdot(1,-1,0)=(0,1,0)+\frac{1}{2}(1,-1,0)=(\frac{1}{2},\frac{1}{2},0)$
\newline
\newline
$\overrightarrow{v_2}=\frac{1}{\|\overrightarrow{w_2}\|}\overrightarrow{w_2}=\frac{1}{\sqrt{(\frac{1}{2})^2+(\frac{1}{2})^2+0^2}}(\frac{1}{2},\frac{1}{2},0)=\frac{\sqrt{2}}{2}(1,1,0)=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$
\newline Defining $W_2=span\big((\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0),(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0))$
\newline
\newline \textbf{Step3}\newline
$\overrightarrow{w_3}=\overrightarrow{u_3}-proj_{W_2}(\overrightarrow{u_3})=\overrightarrow{u_3}-\langle\overrightarrow{u_3},\overrightarrow{v_1}\rangle\overrightarrow{v_1}-\langle\overrightarrow{u_3},\overrightarrow{v_2}\rangle\overrightarrow{v_2}=(2,3,1)-(2\cdot\frac{1}{\sqrt{2}}++3\cdot(-\frac{1}{\sqrt{2}})+1\cdot0)(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)-(2\cdot\frac{1}{\sqrt{2}}+3\cdot\frac{1}{\sqrt{2}}+1\cdot0)(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)=(2,3,1)+\frac{1}{\sqrt{2}}(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)-\frac{5}{\sqrt{2}}(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)=(0,0,1)$
\newline
\newline
$\overrightarrow{v_3}=\frac{1}{\|\overrightarrow{w_3}\|}\overrightarrow{w_3}=\frac{1}{\sqrt{0^2+0^2+1^2}}(0,0,1)=(0,0,1)$
\newline
\newline
\newline
Finally, the orthogonal basis for $\mathbb{R}^3$ obtained from given basis S=$\big((1,-1,0),(0,1,0),(2,3,1))$ is $$\big((\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0),(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0),(0,0,1))$$ \newline
As we can clearly see, this set is orthonormal because it contains 3 vectors(and dimension of V equal to 3), it is orthogonal(real inner product of any two vectors in it is equal to 0) and norm of any vector in it is equal to 1.
\begin{thebibliography}{10}
\bibitem{1}
Dr. Marcos Alvarez, \emph{Linear algebra, lecture notes} 2011-2012.
\bibitem{2}
Dr. Lipschutz, Dr. Marc Lipson \emph{Shaum's Outlines, Linear Algebra} Third edition, 2000.
\bibitem{3}
Jim Hefferon, Saint Michael's College \href{joshua.smcvt.edu/linearalgebra/book11.pdf}{\emph{Linear Algebra} 2011}.
\bibitem{4}
Wikipedia, The Free Online Encyclopedia \href{en.wikipedia.org/wiki/Inner_product_space}{\emph{Inner Product Space}}.
\bibitem{5}
Wikipedia, The Free Online Encyclopedia: \href{en.wikipedia.org/wiki/Gram-Schmidt_process}{\emph{Gram-Schmidt Process}}.
\end{thebibliography}
\end{document}


А так, на сим заканчивается учебный год тчк

@темы: Психи из мат. кафедры., Ха ха, Доктор... это лечится?, студенческая жизнь

URL
Комментарии
2012-06-22 в 22:10 

Второе, однозначно. )

URL
2012-06-22 в 23:51 

New Jack
Discover. Acquire. Master. EVOLVE. Repeat
Гость,
Да, походу без вариантов :D

URL
2012-06-23 в 23:14 

ANIMAATRA
А так, на сим заканчивается учебный год тчк
Поздравляю)
Я даже спрашивать боюсь, что это такое под катом спрятано=)

2012-06-25 в 01:50 

New Jack
Discover. Acquire. Master. EVOLVE. Repeat
ANIMAATRA,
Спасибо)
Под катом моя курсовая. Всего-лишь команды для текстового редактора. То есть, стоит это вбить в одну программу и оно выдаст .pdf фаил с текстом. А зачем такой геморой? Просто на ворде, например, очень неудобно работать с математикой. Тут это проще.

URL
2012-06-25 в 22:00 

ANIMAATRA
Увы, ссылка не открывается, но я верю тебе на слово. Боюсь, вникать в точные науки уже смертельно опасно для моих филологических мозгов)))

     

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