What is wrong in my code, I cant solve it

Question

@article{wikipedia,
title = "Scripting language",
author = "[1]",
url = {http://en.wikipedia.org/wiki/Scripting_language},
}
@book{kailath1980linear,
  title={Linear Systems},
  author={Kailath, T.},
  isbn={9780135369616},
  lccn={79014928},
  series={Information and System Sciences Series},
  year={1980},
  publisher={Prentice-Hall}
}

here is my bib file and here is my latex

\section{Introduction}

The project will consider direct methods for solving linear systems of equations. A number of methods will be investigated. The project will examine methods for solving general linear systems including Gaussian elimination and LU factorisation as well as methods for special linear systems such as the Cholesky and LDL decompositions. 

\section{Structure of the thesis}
In this project, we will start from the analysis of linear system using the linear equations and matrix, and then we will enter the general linear system we will start from the first semester this include the method of Gaussian  elimination and the LU method  also part the end of the semester we will have completed the first two parts and also start working on the third part third is the method chelosky LDL, in the end, at this three-part there will have been  completed the report as well include the final presentation of the project.

\section{Aims of the Project}
In this project I will gain on linear systems how they work in real life. I will be examined in methods of resolving tables and various other methods. 
\chapter{PROJECT SCOPE}
\section{Brief definition  of a linear system}

Linear systems are systems of equations in which the variables are never multiplied with each other but only with constants and then summed up.A Linear systems,also it's a set of collection of linear equations with the same unknowns,which we're trying to define so that they verify all the equations. For example $n$ linear equations in $n$ unknowns $x_1,x_2,...,x_n$.
We also we will use the formula, 
$$\sum_{j=1}^n a_{i,j}x_j=b_i,\quad i=1,2,\ldots,n.\cite{poole2005linear}$$


\section{A Historical Background about linear equation}
First we will start to tell some of the important information and dates concerning linear systems.Starting first I will refer to the year in which the first linear has written 300 of the ancient Babylonians who solved 2 equation with 2 unknowns.The Chinese are between 200 with 100 used matrix.The method used is essentially the Gauss method.The Cardan in the book of Ars Magna at 1545 it's been invented the cramer's rules to resolve 2 equations, approaching the concept of determinants.The Cramer to 1750 gives the general rule that he is now known by his name Cramer's rules  for the one solution 
nxn systems. 
\section{Number of solution in Linear equation}
In linear systems can exist two types of solutions.A solution of a linear system is an assignment of values to the variables\quad$x_1,x_2 \ldots x_n$ such that each of the equations is satisfied with type of solution set.
In general a linear system may have:
   \begin{enumerate}
    \item infinitely solution,
    \item unique solution,
    \item no solution 
    \end{enumerate}
\section{Geometric interpretation}
For a linear system equation has to variable $x$ and $y$ each then determinate a
line.Because is a solution to a linear system must satisfy all of the equations,
the solution set is the intersection of these lines, and is hence either a line, a
single point, or the empty set.
For example,in the Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns.Have an example to illustrate, \label{geometriacal example} 
\begin{figure}[htp]
\centering
\includegraphics[scale=.6]{pics/geogebra-export}
\caption{geometrical example}
\label{fig:illustration}
\end{figure}
\subsection{Cramer's rule}
In linear algebra, Cramer's rule is an  formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the linear system has a unique solution.Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.This considers a system of two linear equations in two variables.

Why Cramer's rule is not suitable for solving large linear systems?\\
This is a good question the solution through the rules of the Cramer nation is impossible because the time required to solve a linear equation is long there is no solution in Cramer's rule in large linear systems.
\section{Numerical method} 
The Linear System also can be applied with  method of solution: 
 \begin{itemize}
 \item Direct method 
 \item iterative method 
 \end{itemize}

\paragraph{Direct method}
\vspace{0.8cm}
A direct method are the results in linear systems can be counted after the experience and many operations.In the absence of a path leads to the exact solution of the linear system equation.In practice however, the accumulation of rounding errors combined with possible algorithm instability or system function, using the method can lead to a completely useless solution. Essentially, all  direct methods used are variants of the Gauss elimination method.
\textbf{Iterative method} 

$x_{1} \quad x_{2} \quad .... \in\mathbb{R}^n$.In the project will not use this method we use the direct method as well. 

\cite{wikipedia}

\chapter{General Linear Systems}
\section{Gaussian Elimination}
Gaussian Elimination also known as 
Gaussian elimination works following this step 
First Steps:

\bibliographystyle{plain}
\bibliography{bibliography}
\end{document}

the error u I can't see my bibliography on my pdf and he said cite ..... on page 5 is undefined which is wrong.

Answer 1

You asked,

What is wrong in my code?

Some observations:

• One cannot be entirely sure what all may be wrong, as you don't specify which document class you employ (report? book? something else?) and which packages you load. I gather that, at a minimum, you load the following packages: amsmath, amssymb, graphicx. You may also be loading a citation management package (cite?, natbib?) and either url or xurl -- along with other packages.

• You also don't specify what exactly you do. After an initial LaTeX run, did you run BibTeX, followed by two more LaTeX runs? If you ran BibTeX, did you get any warning or error messages?

• The document features the instruction \cite{poole2005linear}, but there's no entry with key poole2005linear among the bibliographic entries.

• Not an outright error but possibly a symptom of other problems: The bib entry kailath1980linear is not used.

• It's wrong to use the entry type @article for the wikipedia entry. The @article entry type should only ever be used for pieces published in academic journals. Do look into the entry type @misc. The author field of the wikipedia entry must surely be wrong; do give author = "Wikipedia" a try.

• The instruction \vspace{0.8cm} has no effect whatsoever if encountered immediately after a \paragraph instruction. I have no idea why \vspace{0.8cm} is there. If you need a line break after the sectioning header, don't use \paragraph (or, for that matter, \subparagraph).

• Don't use $$ to initiate and terminate display-math mode in a LaTeX document. Instead, use \[ and \]. For more information on why you shouldn't be using $$, see the posting Why is \[ ... \] preferable to $$ ... $$?

• The instruction \label{geometriacal example} seems out of place? What typographic object are you trying to label?

• Don't use \\ to force a line break in text mode. Instead, use \par or, better still, create an all-blank line.

• Don't use ..., let alone ...., to create typographic ellipses. Instead, use \dots. And, where appropriate, use commas in a list; thus, don't write $x_1,x_2 \ldots x_n$; instead, please write $x_1,x_2,\ldots,x_n$.

Answer 2

ok thank you for all of your answer .Put the problem is there .the ref can not desplay in my pdf i sent it to my profesor and he said is ok put you can not compile two or more times the latex put now i try this and the problem is here i cant resovled the problem i dont now why.

   @book{atkinson1978,
  title={An introduction to numerical analysis},
  author={Atkinson, K.E.},
  isbn={9780471029854},
  lccn={78006706},
  year={1978},
  publisher={Wiley}
}

@book{poole2005,
  title={Linear Algebra: A Modern Introduction},
  author={Poole, D.},
  isbn={9780534998455},
  lccn={2004111976},
  year={2005},
  publisher={Thomson Brooks/Cole}
}
\chapter{Background of the Project}

\section{Introduction}
I will describe an introduction about a linear systems equation\newline Linear systems are systems of equations in which the variables are never multiplied with each other but only with constants and then summed up.A Linear systems,also it's a set of collection of linear equations with the same unknowns,which we're trying to define so that they verify all the equations. For example $n$ linear equations in $n$ unknowns $x_1,x_2,...,x_n$.
We also we will use the formula, 
$$\sum_{j=1}^n a_{i,j}x_j=b_i,\quad i=1,2,\ldots,n.\cite{atkinson1978}$$
\section{A Historical Background about linear equation}
First we will start to tell some of the important information and dates concerning linear systems.Starting first I will refer to the year in which the first linear has written 300 of the ancient Babylonians who solved 2 equation with 2 unknowns.The Chinese are between 200 with 100 used matrix.The method used is essentially the Gauss method.The Cardan in the book of Ars Magna at 1545 it's been invented the cramer's rules to resolve 2 equations, approaching the concept of determinants.The Cramer to 1750 gives the general rule that he is now known by his name Cramer's rules  for the one solution or a no solution in a linear equation.  
\section{Number of solution in Linear equation}
In linear systems can exist two types of solutions.A solution of a linear system is an assignment of values to the variables\quad$x_1,x_2 \ldots x_n$ such that each of the equations is satisfied with type of solution set.
In general a linear system may have:
   \begin{enumerate}
    \item infinitely solution,
    \item unique solution,
    \item no solution 
    \end{enumerate}
\section{Geometric interpretation}
For a linear system equation has to variable $x$ and $y$ each then determinate a
line.Because is a solution to a linear system must satisfy all of the equations,
the solution set is the intersection of these lines, and is hence either a line, a
single point, or the empty set.\newline
For instance,in the Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns.Have an example to illustrate, 
\label{geometriacal example}
\begin{figure}[htp]
\centering
\includegraphics[scale=1]{images/geogebra-export}
\caption{geometrical example}
\label{fig:illustration}
\end{figure}
\section{Cramer's rule}

In linear algebra,Cramer's rule is an  formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the linear system has a unique solution.Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.This consider a system of two linear equations in two variables.

Why cramer's rule is not suitable for solving large linear system?\newline 
This is a good question the solution through the rules of the cramer nation is impossible because the time required to solve a linear equations is long there is no solution in cramer's rule in larg linear system.

\section{Method for solving linear Systems equation}
\par Direct Method 

A direct method are the results in linear systems can be count after the experience and many operation.In the absence of a path leads to the exact solution of the linear system equation.In practice however, the accumulation of rounding errors combined with possible algorithm instability or system function, using the method can lead to a completely useless solution. Essentially, all the direct methods used are variants of the Gauss elimination method.

\par iterative method  

$x_{1} \quad x_{2} \quad .... \in\mathbb{R}^n$.In the following project will not use this method we use the direct method as well for solving linear systems equation.