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Matrix Theory:
Linear Algebra and Beyonds
Peter Tan
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Front Matter
1
System of linear equations
1.1
Three elementary row operations
1.1.1
How to input a regular matrix
1.1.2
How to input a matrix with variables
1.1.3
Three elementary row operations
1.2
Reduced row echelon form
1.3
General solutions
1.4
Matrix Form
1.5
Exercise
2
Matrix Operations
2.1
Vector
2.1.1
row and column vectors
2.1.1.1
How to get a submatrix
2.2
Matrix Multiplication
2.2.1
Matrix Multiplication
2.2.2
\(AB\neq BA\)
2.3
Inverse Matrix
2.4
Elementary matrix
3
Determinant
3.1
Determinant
3.1.1
Fast way to get the determinant
3.1.2
The slow one
3.2
Cramer’s Rule
4
Vector Space
4.1
Visualization of vectors
4.2
Linear independent
4.3
Span
4.4
Row Space and Column Space of a Matrix
5
Inner Product Space
5.1
Euclidean Space
5.1.1
Dot product in
\(\mathbb{R}^{n}\)
5.1.2
Length
5.1.3
Angle
5.1.4
Distance
5.1.5
Cross Product
5.2
Orthogonal Projection
5.2.1
Orthogonal and orthonormal set
5.2.2
Orthogonal Projection
5.3
Gram-Schmidt Process
5.4
Inner product space
5.4.1
Abstract Vector Space
5.4.2
Inner Product Space
5.4.3
Gram-Schmidt Process in an Inner Product Space
6
Linear Transformation
6.1
The matrix of a Linear map
6.1.1
Linear Transformation
6.1.2
Linear Transformation
\(T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\)
6.2
The Kernel and range of
\(T\)
6.3
Understand the linear transformation
6.3.1
Matrix relative to bases
7
Eigen-theory
7.1
Eigenvalue and Eigenvector
7.1.1
Eigenvalue of a square matrix
7.1.2
Characteristic Polynomial
7.1.3
Algebraic and Geometric Multiplicities
7.2
Diagonalization
7.3
Orthogonally Diagonalization
7.4
Application: Find
\(e^{At}\)
Backmatter
Chapter
6
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.
6.1
The matrix of a Linear map
6.2
The Kernel and range of
\(T\)
6.3
Understand the linear transformation
