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Linear Algebra with SageMath
Learn math with open-source software
Allaoua Boughrira, Hellen Colman, Michael Kattner, Samuel Lubliner
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Front Matter
Preface
Acknowledgements
From the Student Authors
Authors and Contributors
Colophon
1
Getting Started
1.1
Intro to Sage
1.1.1
Sage as a Calculator
1.1.2
Variables and Names
1.2
Data Types
1.3
Flow Control Structures
1.3.1
Conditional Statements
1.3.1.1
Exception Handling
1.3.2
Iteration
1.3.2.1
List Comprehension
1.3.3
Other Flow Control Structures
1.4
Defining Functions
1.5
Object-Oriented Programming
1.5.1
Objects in Sage
1.5.2
Dot Notation and Methods
1.5.3
Sage’s Set Class
1.6
Display Values
1.7
Debugging
1.8
Documentation
1.9
Miscellaneous Features
1.9.1
Reading and Writing Files in Sage
1.9.2
Executing Shell Commands in Sage
1.9.3
Importing and Exporting Data (CSV, JSON, TXT)
1.9.4
Using External Libraries in Sage
1.10
Run Sage in the browser
2
Vectors and Matrices: The Basics
2.1
Vectors
2.2
Matrices
3
System of Equations
3.1
Solving Equations
3.2
Solving Systems of Equations
3.3
Graphing of Systems
4
System of Equations with Matrices
4.1
Gauss-Jordan Elimination
4.1.1
Augmented Matrix
4.1.2
RREF
4.2
Pivots
4.2.1
Compatible Unique Solutions (Case I)
4.2.2
Compatible Infinitely Many Solutions (Case II)
4.2.3
Incompatible (Case III)
4.3
Matrix Equations
4.3.1
Case I
4.3.2
Case II
4.3.3
Case III
4.4
LU Decomposition
4.4.1
Assisted Calculation of LU factorization
4.4.2
Sage Calculation of LU factorization
4.4.3
Solving a system of equations using LU
5
Vectors
5.1
Basic Arithmetic Operations
5.2
Dot and Cross Products
6
Matrices
6.1
Special Matrices
6.2
Operations With Matrices
6.3
Transpose and Conjugate
6.4
Trace and Norm
7
Determinants
7.1
Determinants
7.2
Cramer’s Rule
8
Adjugate Matrix
8.1
Adjugate Matrix
8.2
Minors of a Matrix
8.3
Cofactors Matrix
8.4
Alternative Minors/Cofactors Computation
8.5
Historical Note: Adjoint vs. Adjugate
9
Inverse Matrix
9.1
Inverse Matrix
9.1.1
Echelon Method
9.1.2
Adjugate Method
9.1.3
Comparing Methods
10
Vector Spaces
10.1
Definition of Vector Spaces
10.2
Subspaces
10.2.1
Spans
10.2.2
Linear Independence
10.3
Bases
10.3.1
Extracting a Basis from a Generator
10.3.2
Coordinates in a Basis
10.3.3
Change of Basis
10.3.4
Change of Basis Matrix
10.4
Null Spaces
10.4.1
Manual Calculation of the Kernel
10.4.2
Kernel Method
10.5
Gram-Schmidt Process
10.5.1
Finding an Orthogonal Basis
10.5.2
Finding the Orthonormal Basis
11
Eigenvectors and Eigenvalues
11.1
Definition
11.2
Eigenvalues
11.2.1
Assisted Calculation of Eigenvalues
11.2.2
Sage Calculation of Eigenvalues
11.3
Eigenvectors
11.3.1
Manual Calculation of Eigenvectors
11.3.2
Sage Calculation of Eigenvectors
12
Diagonalization
12.1
Diagonalization of a Matrix
13
Linear Transformations
13.1
Definition
13.1.1
Sage Calculation
13.2
Kernel and Image
13.3
Change of Basis
Back Matter
References
Index
Chapter
3
System of Equations
Linear systems of equations are often given geometric meaning as intersections between lines, planes, or higher dimensional equivalents. This chapter will introduce how to use Sage to find solutions of equations in their familiar form.
🔗
3.1
Solving Equations
3.2
Solving Systems of Equations
3.3
Graphing of Systems
🔗
