Introduction

Course Objectives

In this course, we will learn the basics of Linear Algebra by being hands-on using the open source SageMath software. Only basic math knowledge required to follow this course!!

I created this workshop as revision notes while I was learning Linear Algebra, but it may be useful for students who prefer to get hands-on with concepts before jumping deep into theory.

If you like this project, please give it a star on Github.

How to run the notebooks

View: to view the notebooks, you can start here

Interact: If you would like an interactive environment to experiment with the notebook code, you can run then online for free here

Notes for the interactive environment:

• it may take about 5 mins to launch the environment

• notebook state is not persisted - anything you add is lost when the notebook session closes

• the notebooks have an inactivity timeout of 10 mins after which the session closes

Feedback

If after following this workshop you feel that the objectives haven’t been achieved, please email me with your feedback.

Pre-requisites

You should have basic Python knowledge and be familiar with using Jupyter Notebooks. If not, the following free hands-on workshops are recommended:

• Python 101

• Jupyter Notebooks 101

Contents

Notebook	Description
01 Vectors	Learn the fundamentals of working with vectors, including vector addition, scalar multiplication, and vector operations in geometric and algebraic contexts.
02 Matrices	Explore matrices, their properties, and operations such as addition, multiplication, and inversion. Understand how matrices are used to represent linear transformations.
03 Gauss Method	Learn about the Gaussian elimination method for solving systems of linear equations and transforming matrices into reduced row-echelon form.
04 Solution Sets	Investigate solution sets of linear systems, including unique solutions, infinite solutions and no solutions.
05 Vector Spaces	Study vector spaces, their properties, and examples from various fields. Explore concepts such as linear independence, spanning sets, and basis vectors.
06 Linear Dependence	Understand the concept of linear dependence and independence among vectors and its implications for vector spaces and systems of equations.
07 Span	Study the span of a set of vectors, which represents all possible linear combinations of those vectors. Understand its connection to subspaces and basis vectors.
08 Basis	Learn about basis vectors, which form the building blocks for vector spaces. Understand how basis vectors can be used to represent other vectors uniquely.
09 Subspaces	Explore subspaces of vector spaces, including their definitions, properties, and examples, such as column spaces, row spaces, and null spaces.
10 Homogenous Linear Systems	Investigate homogeneous linear systems, which have solutions that form vector spaces. Understand their relationship to null spaces and the nullity of matrices.
11 MatrixSpaces	Explore spaces of matrices, including their properties, dimensions, and relationships to vector spaces.
12 Linear Transformations	Study linear transformations, which are functions that preserve vector addition and scalar multiplication. Understand their properties and geometric interpretations.
13 Null Spaces and Column Spaces	Investigate the relationship between the null space and the column space of a matrix, as well as their dimensions and properties.
14 Projections	Explore projections of vectors onto subspaces, including orthogonal and non-orthogonal projections. Understand their applications in various fields.
15 Orthogonality	Study orthogonality between vectors and subspaces, including orthogonal complements and orthogonal bases. Understand its significance in geometry and linear algebra.
16 Determinants	Define determinants, explain their geometric interpretation, and show how to compute them. Understand their role in determining properties of matrices and their applications.
17 Eigenvalues and Eigenvectors	Introduce the concept of eigenvalues and eigenvectors, their geometric interpretation, and their computation. Understand their significance in diagonalizing matrices and solving systems of differential equations.
18 Matrix Decomposition	LU Decomposition, QR Decomposition, Cholesky Decomposition, Eigendecomposition
19 Singular Value Decomposition	Singular Values, Singular Vectors, Applications of SVD, Principal Component Analysis
98 Leftovers	Least Squares, Distance, Transpose.
99 Quickref	SageMath linear algebra quick reference, and tutorials.

License

Source

snowch/learn_linear_algebra