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<h1 id="computational-mechanics-4---linear-algebra">Computational Mechanics 4 - Linear Algebra</h1>
<p>Welcome to Computational Mechanics Module #4! In this module we will explore applied linear algebra for engineering problems and revisit the topic of linear regression with a new toolbox of linear algebra. Our main goal, is to transform large systems of equations into manageable engineering solutions.</p>
<p><a href="./notebooks/01_Linear-Algebra.ipynb">01_Linear-Algebra</a></p>
<ul>
<li>How to solve a linear algebra problem with <code>np.linalg.solve</code></li>
<li>Creating a linear system of equations</li>
<li>Identify constants in a linear system <span class="math inline">\(\mathbf{A}\)</span> and <span class="math inline">\(\mathbf{b}\)</span></li>
<li>Identify unknown variables in a linear system <span class="math inline">\(\mathbf{x}\)</span></li>
<li>Identify a <strong>singular</strong> or <strong>ill-conditioned</strong> matrix</li>
<li>Calculate the <strong>condition</strong> of a matrix</li>
<li>Estimate the error in the solution based upon the condition of a matrix</li>
</ul>
<p><a href="02_Gauss_elimination.ipynb">02_Gauss_elimination</a></p>
<ul>
<li>Graph 2D and 3D linear algebra problems to identify a solution (intersections</li>
<li>of lines and planes)</li>
<li>How to solve a linear algebra problem using <strong>Gaussian elimination</strong> (<code>GaussNaive</code>)</li>
<li>Store a matrix with an efficient structure <strong>LU decomposition</strong> where <span class="math inline">\(\mathbf{A=LU}\)</span></li>
<li>Solve for <span class="math inline">\(\mathbf{x}\)</span> using forward and backward substitution (<code>solveLU</code>)</li>
<li>Create the <strong>LU Decomposition</strong> using the Naive Gaussian elimination process (<code>LUNaive</code>)</li>
<li>Why partial <strong>pivoting</strong> is necessary in solving linear algebra problems</li>
<li>How to use the existing <code>scipy.linalg.lu</code> to create the <strong>PLU decomposition</strong></li>
<li>How to use the <strong>PLU</strong> efficient structure to solve our linear algebra problem (<code>solveLU</code>)</li>
</ul>
<p><a href="03_Linear-regression-algebra.ipynb">03_Linear-regression-algebra</a></p>
<ul>
<li>How to use the <em>general least squares regression</em> method for almost any function</li>
<li>How to calculate the coefficient of determination and correlation coefficient for a general least squares regression, <span class="math inline">\(r^2~ and~ r\)</span></li>
<li>How to plot and read a <strong>training-testing</strong> plot</li>
<li>How to divide data into <strong>training</strong> and <strong>testing</strong> data for analysis</li>
<li>Why we need to avoid <strong>overfitting</strong></li>
<li>How to construct general least squares regression using the dependent and independent data to form <span class="math inline">\(\mathbf{y}=\mathbf{Za}\)</span>.</li>
<li>How to construct a piecewise linear regression</li>
</ul>
</body>
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	<body>
	<h1 id="computational-mechanics-4---linear-algebra">Computational Mechanics 4 - Linear Algebra</h1>
	<p>Welcome to Computational Mechanics Module #4! In this module we will explore applied linear algebra for engineering problems and revisit the topic of linear regression with a new toolbox of linear algebra. Our main goal, is to transform large systems of equations into manageable engineering solutions.</p>
	<p><a href="./notebooks/01_Linear-Algebra.ipynb">01_Linear-Algebra</a></p>
	<ul>
	<li>How to solve a linear algebra problem with <code>np.linalg.solve</code></li>
	<li>Creating a linear system of equations</li>
	<li>Identify constants in a linear system <span class="math inline">\(\mathbf{A}\)</span> and <span class="math inline">\(\mathbf{b}\)</span></li>
	<li>Identify unknown variables in a linear system <span class="math inline">\(\mathbf{x}\)</span></li>
	<li>Identify a <strong>singular</strong> or <strong>ill-conditioned</strong> matrix</li>
	<li>Calculate the <strong>condition</strong> of a matrix</li>
	<li>Estimate the error in the solution based upon the condition of a matrix</li>
	</ul>
	<p><a href="02_Gauss_elimination.ipynb">02_Gauss_elimination</a></p>
	<ul>
	<li>Graph 2D and 3D linear algebra problems to identify a solution (intersections</li>
	<li>of lines and planes)</li>
	<li>How to solve a linear algebra problem using <strong>Gaussian elimination</strong> (<code>GaussNaive</code>)</li>
	<li>Store a matrix with an efficient structure <strong>LU decomposition</strong> where <span class="math inline">\(\mathbf{A=LU}\)</span></li>
	<li>Solve for <span class="math inline">\(\mathbf{x}\)</span> using forward and backward substitution (<code>solveLU</code>)</li>
	<li>Create the <strong>LU Decomposition</strong> using the Naive Gaussian elimination process (<code>LUNaive</code>)</li>
	<li>Why partial <strong>pivoting</strong> is necessary in solving linear algebra problems</li>
	<li>How to use the existing <code>scipy.linalg.lu</code> to create the <strong>PLU decomposition</strong></li>
	<li>How to use the <strong>PLU</strong> efficient structure to solve our linear algebra problem (<code>solveLU</code>)</li>
	</ul>
	<p><a href="03_Linear-regression-algebra.ipynb">03_Linear-regression-algebra</a></p>
	<ul>
	<li>How to use the <em>general least squares regression</em> method for almost any function</li>
	<li>How to calculate the coefficient of determination and correlation coefficient for a general least squares regression, <span class="math inline">\(r^2~ and~ r\)</span></li>
	<li>How to plot and read a <strong>training-testing</strong> plot</li>
	<li>How to divide data into <strong>training</strong> and <strong>testing</strong> data for analysis</li>
	<li>Why we need to avoid <strong>overfitting</strong></li>
	<li>How to construct general least squares regression using the dependent and independent data to form <span class="math inline">\(\mathbf{y}=\mathbf{Za}\)</span>.</li>
	<li>How to construct a piecewise linear regression</li>
	</ul>
	</body>
	</html>