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Linear Algebra Lab Manual:
Exploring Linear ALgebra with SageMath
Yilan Tan
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Front Matter
1
Introduction to SageMath
1.1
How to use this lab Manual
1.2
What is Sagemath
1.3
Using Sage as a Calculator
1.4
Using Sage with Common Functions
1.5
Using Sage to solve equations
1.5.1
Math Features
1.6
Assignment #1
2
Gaussian-Jordan Elimination
2.1
How to input a matrix into SageMath
2.1.1
How to input a regular matrix
2.1.2
How to input an irregular matrix
2.2
Three elementary row operations
2.3
Reduced Echelon form
2.4
Chanllenge Question
3
Matrix Operations
3.1
The row vectors, column vectors and submatrix
3.1.1
row and column vectors
3.1.2
How to get a submatrix
3.1.3
Constructing Matrix from vectors
3.2
Matrix Multiplication
3.2.1
Matrix Multiplication
3.2.2
\(AB\neq BA\)
3.2.3
The
\(n\)
-th Power of a matrix
3.3
Inverse Matrix
3.4
Elementary matrix
3.5
Assignment
4
Determinant
4.1
Determinant in SageMath
4.2
Recursive Method
4.3
Cramer’s Rule
5
Vector Space
5.1
Visualization of vectors
5.2
Linear independent
5.3
Span
5.4
Row Space and Column Space of a Matrix
6
Inner Product Space
6.1
Euclidean Space
6.1.1
Dot product in
\(\mathbb{R}^{n}\)
6.1.2
Length
6.1.3
Angle
6.1.4
Distance
6.1.5
Cross Product
6.2
Orthogonal Projection
6.2.1
Orthogonal and orthonormal set
6.2.2
Orthogonal Projection
6.3
Gram-Schmidt Process
6.4
Inner product space
6.4.1
Abstract Vector Space
6.4.2
Inner Product Space
6.4.3
Gram-Schmidt Process in an Inner Product Space
7
Linear Transformation
7.1
The matrix of a Linear map
7.1.1
Linear Transformation
7.1.2
Linear Transformation
\(T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\)
7.2
The Kernel and range of
\(T\)
7.3
Understand the linear transformation
7.3.1
Matrix relative to bases
8
Eigen-theory
8.1
Eigenvalue and Eigenvector
8.1.1
Eigenvalue of a square matrix
8.1.2
Characteristic Polynomial
8.1.3
Algebraic and Geometric Multiplicities
8.2
Diagonalization
8.3
Orthogonally Diagonalization
8.4
Application: Find
\(e^{At}\)
Backmatter
Chapter
7
Linear Transformation
Motivation
The properties of a linear transformation are totally encapsulated by a corresponding matrix. The matrix is much easier to understand than the linear transformation.
7.1
The matrix of a Linear map
7.2
The Kernel and range of
\(T\)
7.3
Understand the linear transformation
