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@@ -3,6 +3,8 @@ title: "About" | |
| date: | ||
| license: false | ||
| comments: false | ||
| latex: true | ||
| toc: false | ||
| --- | ||
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| # About | ||
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| --- | ||
| title: "Calculus 1 (MTH-251)" | ||
| date: 2023-08-13T07:29:02-07:00 | ||
| draft: false | ||
| --- | ||
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| # Calculus 1 (MTH-251) Notes |
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| --- | ||
| title: "Applied Linear Algebra 1 (MTH-261)" | ||
| draft: false | ||
| comments: false | ||
| --- | ||
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| # Applied Linear Algebra 1 (MTH-261) Notes | ||
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| <ul> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/full">Full Notes</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/introduction">0 Introduction</a></li> | ||
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| <li style="list-style-type: none;"> | ||
| <a href="/notes/mth-261/lec-01">Lecture 1: Linear Systems</a> | ||
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| <ul> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-01/01">1.1 Introduction to Linear Systems</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-01/02">1.2 Basic Algebra Review</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-01/03">1.3 Introduction to Matrices</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-01/04">1.4 Solving Linear Systems with Row Operations</a></li> | ||
| </ul> | ||
| </li> | ||
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| <li style="list-style-type: none;"> | ||
| <a href="/notes/mth-261/lec-02">Lecture 2: Row Reduction and Echelon Forms</a> | ||
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| <ul> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-02/01">2.1 Pivot Position</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-02/02">2.2 The Row Reduction Algorithm</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-02/03">2.3 General Solutions of Linear Systems</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-02/04">2.4 Criteria for Consistent Linear Systems</a></li> | ||
| <li style="list-style-type: none; margin-left: 50px"><a href="/notes/mth-261/lec-02/05">2.5 Some Applications</a></li> | ||
| </ul> | ||
| </li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-03">Lecture 3: Vectors and Vector Equations</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-04">Lecture 4: Matrix Equations</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-05">Lecture 5: Solution Sets and More Application</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-06">Lecture 6: Linear Independence</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-07">Lecture 7: Intro to Linear Transformations</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-08">Lecture 8: Matrix of Linear Transformations</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-09">Lecture 9: Matrix Operations and Inverses</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-10">Lecture 10: Determinants</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-11">Lecture 11: Crammer's Rule</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-12">Lecture 12: Subspaces</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-13">Lecture 13: Eigenvalues and Eigenvectors</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-14">Lecture 14: Diagonalization</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-15">Lecture 15: Diagonalization</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-16">Lecture 16: Orthogonality</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-17">Lecture 17: More Projections and the Gram-Schmidt Process</a></li> | ||
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| <li style="list-style-type: none;"><a href="/notes/mth-261/lec-18">Lecture 18: Least-Squares Problems</a></li> | ||
| </ul> |
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| --- | ||
| title: "0 Introduction" | ||
| draft: false | ||
| comments: false | ||
| --- | ||
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| # 0 Introduction | ||
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| {{< endNote nextNoteDisplay="Lecture 1" nextNoteLink="/lec-01/" >}} |
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| --- | ||
| title: "1.1 Introduction to Linear Systems" | ||
| draft: false | ||
| comments: false | ||
| latex: true | ||
| --- | ||
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| # 1.1 Introduction to Linear Systems | ||
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| {{< definition name="Linear Equation" counter="1" >}} | ||
| <div> | ||
| A \(\textbf{linear equation}\) in the variables \(x_1, x_2, \dots, x_n\) is | ||
| an equation that can be written in the form \(c_1x_1 + c_2x_2 + \dots + | ||
| c_nx_n = k\), where \(n \in \R\) and \(k, c_1, c_2, \dots, c_n \in \R \cup | ||
| \mathbb{C}\). | ||
| </div> | ||
| {{< /definition >}} | ||
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| {{< example >}} | ||
| Below are some examples of linear equations. | ||
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| \begin{multicols}{2}\noindent | ||
| \begin{enumerate} | ||
| \label{enum:linear_equation_1} | ||
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| \item $4.6x_1 + 2x_2 - 5x_3 - x_4 = 1.2$ | ||
| \item $x_2 = 2(\sqrt{6} - x_1) + x_3$ | ||
| \columnbreak | ||
| \item $9x_1 - 5x_2 + 2 = x_2$ | ||
| \item $x_1 - x_3 + 4x_5 = 1$ | ||
| \end{enumerate} | ||
| \end{multicols} | ||
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| Here are some equations that aren't linear. | ||
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| \begin{multicols}{2}\noindent | ||
| \begin{enumerate} | ||
| \label{enum:non_linear_equation_1} | ||
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| \item $\incorrect{x_1 \cdot x_2} = 5 + x_1$. | ||
| \item $\incorrect{\sfrac{x_2}{x_1}} + x_3 = 30$. | ||
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| \item $\incorrect{x_1^2} + 8x_3 = x_1$ | ||
| \item $\incorrect{\sqrt{x_1}} + x_2 = 120$ | ||
| \end{enumerate} | ||
| \end{multicols} | ||
| {{< /example >}} | ||
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| \begin{definition}[System of Linear Equations] | ||
| \label{def:system_of_linear_equations} | ||
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| A \textbf{system of linear equations} or \textbf{linear system} is a | ||
| collection of one or more linear equations with them same set of variables | ||
| $x_1, x_2, \dots, x_n$. | ||
| \end{definition} | ||
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| \begin{example} | ||
| \label{exm:system_of_linear_equations} | ||
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| Below are some examples of linear systems. | ||
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| \begin{multicols}{3}\noindent\centering | ||
| \[% | ||
| \sysdelim..\systeme{ | ||
| 5x_1 + x_2 = 1, | ||
| -4x_2 + 8x_3 = 2, | ||
| x_1 - 3x_3 = 0 | ||
| } | ||
| .\]% | ||
| $3$ equations, $3$ variables\columnbreak | ||
| \columnbreak | ||
| \[% | ||
| \sysdelim..\systeme{ | ||
| x_1 + x_2 - 2x_4 = 12, | ||
| 4x_3 + 6x_4 = -2, | ||
| 2x_1 - x_2 + 9x_3 = 10 | ||
| } | ||
| .\]% | ||
| $3$ equations, $4$ variables\columnbreak | ||
| \[% | ||
| \sysdelim..\systeme{ | ||
| 9x_1 - x_2 = 0, | ||
| 7x_1 = 1, | ||
| -6x_2 = 3 | ||
| } | ||
| .\]% | ||
| $3$ equations, $2$ variables | ||
| \end{multicols} | ||
| \end{example} | ||
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| \begin{definition}[Solution of Linear System] | ||
| \label{def:solution_of_linear_system} | ||
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| A \textbf{solution} of a linear system is a list $\left\{s_1, s_2, \dots, | ||
| s_n\right\}$ of numbers that makes each equation a true statement when the | ||
| values of $s_1, \dots, s_n$ are substituted for $x_1, \dots, x_n$. | ||
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| The set of all possible solutions is called the \textbf{solution set} of the | ||
| linear system. Two linear systems are called \textbf{equivalent} if they have | ||
| the same solution set. That is, $\forall x \in S_1, x \in S_2$, where $S_1$ | ||
| and $S_2$ are solution sets of two linear systems. | ||
| \end{definition} | ||
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| \begin{example} | ||
| \label{exm:solution_of_linear_system} | ||
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| \begin{multicols}{2} | ||
| The list $(5, 6.5,3)$ is a solution to the following linear system | ||
| \[% | ||
| \sysdelim..\systeme{ | ||
| 2x_1 - x_2 + 1.5x_3 = 8, | ||
| -x_1 + 4x_3 = 7 | ||
| } | ||
| ,\]% | ||
| because $2(5) - (6.5) + 1.5(3) \ce 8$ and $-(5) + 4(3) \ce 7$. | ||
| \columnbreak | ||
| \begin{figure}[H] | ||
| \centering | ||
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| \begin{tikzpicture} | ||
| \begin{axis}[ | ||
| xlabel=$x_1$, | ||
| ylabel=$x_2$, | ||
| xmin=-5.25, xmax=6.25, | ||
| ymin=-3.25, ymax=4.25, | ||
| ] | ||
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| \addplot+[domain=-5.25:6.25]{(-1-x)/(-2)} node[below,pos=0.25]{$\ell_1$}; | ||
| \addplot+[domain=-5.25:6.25]{(3+x)/(3)} node[above,pos=0.25]{$\ell_2$}; | ||
| \addplot[soldot] coordinates{(3,2)}; | ||
| \end{axis} | ||
| \end{tikzpicture} | ||
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| \caption{} | ||
| \label{fig:two_lines_intersecting_at_a_point} | ||
| \end{figure} | ||
| \end{multicols} | ||
| \end{example} | ||
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| {{< endNote prevNoteDisplay="Lecture 1" prevNoteLink="/lec-01/" nextNoteDisplay="Section 1.2" nextNoteLink="/lec-01/02/" >}} |
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| --- | ||
| title: "1.2 Basic Algebra Review" | ||
| draft: false | ||
| comments: false | ||
| --- | ||
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| # 1.2 Basic Algebra Review | ||
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| {{< endNote prevNoteDisplay="Section 1.1" prevNoteLink="/lec-01/01/" nextNoteDisplay="Section 1.3" nextNoteLink="/lec-01/03/" >}} |
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| --- | ||
| title: "1.3 Introduction to Matrices" | ||
| draft: false | ||
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| --- | ||
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| # 1.3 Introduction to Matrices | ||
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| {{< endNote prevNoteDisplay="Section 1.2" prevNoteLink="/lec-01/02/" nextNoteDisplay="Section 1.4" nextNoteLink="/lec-01/04/" >}} |
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| title: "1.4 Solving Linear Systems with Row Operations" | ||
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| --- | ||
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| # 1.4 Solving Linear Systems with Row Operations | ||
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| {{< endNote prevNoteDisplay="Section 1.3" prevNoteLink="/lec-01/03/" nextNoteDisplay="Lecture 2" nextNoteLink="/lec-02/" >}} |
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