Engineering Computations Module 4

Engineering Computations is a collection of stackable learning modules, flexible for adoption in different situations. It aims to develop computational skills for students in engineering, but it can also be used by students in other science majors. The course uses the Python programming language and the Jupyter open-source tools for interactive computing.

Module 4: Land on vector spaces

Practical Linear algebra with Python.

Pre-requisite: learning module EngComp 1 of our collection.

This learning module re-imagines the teaching of linear algebra with a visually rich, intuitive approach, enabled by computing with Python. We disregard all the rules-based, memorization-heavy instruction of typical (undergraduate) courses in linear algebra. Instead, we rely on visuals to elucidate the core concepts, and on computational thinking for applying those concepts to useful settings.

Get an interactive session in MyBinder.org with these course materials by clicking on the button below. Select the folder notebooks_en to access the lessons of this course as fully executable Jupyter notebooks.

Lesson 1: Transform all the vectors

What is a vector? The physicist's view versus the computer scientist's view. Fundamental vector operations: visualizing vector addition and multiplication by a scalar. Intuitive presentation of basis vectors, linear combination and span. What is a matrix? A matrix as a linear transformation mapping a vector in one space, to another space. Visualizing linear transformations. Matrix-vector multiplication: a linear combination of the matrix columns. Some special transformations: rotation, shear, scaling. Matrix-matrix multiplication: a composition of two linear transformations. Idea of inverse of a matrix as a transformation that takes vectors back to where they came from.

Lesson 2: The matrix is everywhere

A matrix is a linear transformation… visualize it. Norm of a vector. A matrix maps a circle to an ellipse… visualize it. A vector that doesn't change direction after a linear transformation is an eigenvector of the matrix. A matrix is a system of equations… visualize it (row perspective). Inconsistent and underdetermined systems. A matrix is a change of basis… visualize it. An inverse of that matrix will change the vector's coordinates back to the original basis. Matrices in three-dimensional space: linear transformations in 3D; 3D systems of linear equations; dimension and rank. Visualize the transformations of rank-deficient matrices.

Lesson 3: Eigenvectors for the win

Geometry of eigendecomposition. Eigenvectors revisited: a matrix transforms a circle to an ellipse, whose semimajor and semiminor axes align with the eigenvectors. Composition of scaling transformation and a rotation transformation: not enough! Complete the composition. Symmetric matrices, orthogonal eigenvectors. Eigendecomposition in general. Diagonalizable matrices. Similar matrices. Eigendecomposition is similarity via a change of basis. Compute eigenthings in Python, using NumPy or SymPy. Eigenvalues in ecology: matrix population models. Markov chains. PageRank algorithm.

Lesson 4: Stick to the essentials: SVD

Geometrical interpretation of singular value decomposition (SVD). While eigendecomposition is a combination of change of basis and stretching, SVD is a combination of rotation and stretching, which can be treated as a generalization of eigendecomposition. Example: SVD in image compression. A 2D image can be represented as an array where each pixel is an element of the array. By applying SVD and dropping smaller singular values, we can reconstruct the original image at a lower computational and memory cost. Non-square matrices: SVD in general; pseudo-inverse. Application to linear least squares; linear regression with SVD.

Install the Dependencies

The notebooks are compatible with Python version 3.5 or later, all packages required are listed in environment.yml and requirements.txt. Pick any option below to install the depencies:

Create a conda environment

Use environment.yml to create a conda environment:

conda env create -f environment.yml

This creates a conda environment named "landlinear". For conda 4.6 and later versions, activate this environment with

conda activate landlinear

For conda version prior to 4.6, run source activate landlinear on Linux and MacOS, activate landlinear on Windows.

Install with conda

conda install matplotlib numpy scipy jupyter imageio ipywidgets

Install with pip

Install the packages using requirements.txt for pip:

pip install -r requirements.txt

Check Installation

Run the script check_install.py to check whether the required packages are installed correctly:

python check_install.py