Linear Algebra with Python, Hyun-Seok Son., 2023

Linear Algebra with Python, Hyun-Seok Son, 2023.
    
   This book introduces many basic aspects of linear algebra using Python packages such as numpy and sympy. Chapters 1 and 2 introduce the creation and characteristics of vectors and matrices. These characteristics are advantageous for various characteristics and calculations of functions by connecting linear functions with vectors or matrices. This part is introduced in Chapter 3. The process of converting a function to another function while maintaining its basic characteristics is covered in Chapter 4. This transformation can convert and apply complex functions to a simpler form. This transformation is called decomposition and is introduced in Chapter 5. In this process, various Python functions and packages are applied and their descriptions are attached in the appendix.




Determinant.
The existence of an inverse matrix means that a unique solution set exists in a linear system. In other words, the existence of an inverse matrix for a standard matrix is a necessary condition for having a unique solution set. Conversely, if the inverse matrix of the standard matrix does not exist, various solution sets exist or no solution exists. A matrix that has no inverse is called a singular matrix.

A matrix has the property that if the determinant that can be calculated from the matrix is 0, there is no inverse matrix. In other words, the determinant is the basis for determining whether a matrix is a singular matrix.

Contents.
Chapter 1 Vector & Matrix Contents.
1.1. Vector & Matrix.
1.1.1. Vector.
1.1.2. Matrix.
1.1.3. Index & slicing.
1.1.4. Basic operations.
1.2. Norms and Inequalities.
1.2.1. Vector Norm.
1.2.2. Cauchv-Schwarz & triangle inequality.
1.3. Inner product, cross product & orthogonality.
1.3.1. Inner product.
1.3.2. Meaning of inner product.
1.3.3. Orthogonal vectors.
1.3.4. Outer product.
Chapter 2 Characteristics of Matrix.
Contents.
2.1. Types of Matrices.
2.1.1. Transposed matrix.
2.1.2. Square matrix.
2.1.3. Identity Matrix.
2.1.4. Trace.
2.1.5. Diagonal matrix.
2.1.6. Triangular matrix.
2.1.7. Symmetric matrix.
2.2. Inverse matrix & determinant.
2.2.1. Inverse matrix.
2.2.2. Reduced row echelon formfrref).
2.2.3. Determinant.
Chapter 3 Linear System and Vector Space Contents.
3.1. Linear system.
3.1.1. Linear Combination.
3.1.2. Homoseneous Linear Combination.
3.1.3. Linear independence and linear dependence.
3.2. Vector space and basis.
3.2.1. Subspace.
3.2.2. Basis vector.
3.2.3. Dimension of subspace.
3.2.4. Vector coordinate system.
3.3. Null space and Column space.
3.3.1. Null Space.
3.3.2. Column Space.
3.4. Ranks and Eigenvalue & Eigenvector.
3.4.1. Rank.
3.4.2. Eigenvector and Eigenvalue.
Chapter 4 Transform.
Contents.
4.1. Kernel and Range.
4.2. Linear transformation.
4.3. orthogonality.
4.3.1. Orthogonal vectors.
4.3.2. Orthogonal Set and Linear Combination.
4.3.3. Orthonormal.
4.3.4. Gram-Schmidt process.
4.4. Similarity transformation.
4.4.1. Similarity matrix.
4.4.2. Diagonalization.
Chapter 5 Decomposition.
Contents.
5.1. OR Decomposition.
5.2. Eigenvalue Decomposition.
5.3. Spectral decomposition.
5.3.1. Diagonalization of a symmetric matrix.
5.3.2. Spectral decomposition.
5.4. Quadratic forms.
5.4.1. Quadratic forms.
5.4.2. Sign of quadratic form.
5.4.3. Constrained optimization.
5.5. Singular Value Decomposition.
5.5.1. Eigenvalue decomposition of non-square matrices.
5.5.2. Singular value.
5.5.3. Singular value decomposition.
Appendix Functions.
Index.



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