README.md

# Computational Mechanics 4 - Linear Algebra

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. 

[01_Linear-Algebra](./notebooks/01_Linear-Algebra.ipynb)


* How to solve a linear algebra problem with `np.linalg.solve`
* Creating a linear system of equations
* Identify constants in a linear system $\mathbf{A}$ and $\mathbf{b}$
* Identify unknown variables in a linear system $\mathbf{x}$
* Identify a __singular__ or __ill-conditioned__ matrix
* Calculate the __condition__ of a matrix
* Estimate the error in the solution based upon the condition of a matrix

[02_Gauss_elimination](02_Gauss_elimination.ipynb)


* Graph 2D and 3D linear algebra problems to identify a solution (intersections
* of lines and planes)
* How to solve a linear algebra problem using __Gaussian elimination__ (`GaussNaive`)
* Store a matrix with an efficient structure __LU decomposition__ where  $\mathbf{A=LU}$
* Solve for $\mathbf{x}$ using forward and backward substitution (`solveLU`)
* Create the __LU Decomposition__ using the Naive Gaussian elimination process (`LUNaive`)
* Why partial __pivoting__ is necessary in solving linear algebra problems
* How to use the existing `scipy.linalg.lu` to create the __PLU decomposition__
* How to use the __PLU__ efficient structure to solve our linear algebra problem (`solveLU`)

[03_Linear-regression-algebra](03_Linear-regression-algebra.ipynb)

* How to use the _general least squares regression_ method for almost any function
* How to calculate the coefficient of determination and correlation coefficient for a general least squares regression, $r^2~ and~ r$
* How to plot and read a __training-testing__ plot
* How to divide data into __training__ and __testing__ data for analysis
* Why we need to avoid __overfitting__
* How to construct general least squares regression using the dependent and independent data to form $\mathbf{y}=\mathbf{Za}$. 
* How to construct a piecewise linear regression