머신러닝과 컴퓨터공학에서의 필수과목인 선형대수를 공부해 보고자 한다.
내용은 Gilber strang의 선형대수 목차를 그대로 따라갈 예정이다.
Wish me luck :)
선형대수
- 선형대수(linear algebra)는 벡터 공간, 벡터, 선형 변환, 행렬, 연립 선형 방정식 등을 연구하는 대수학의 한 분야.
- 현대에서는 벡터공간을 주요 연구분야로 함.
- 이와 함께 추상대수학, 함수해석학에도 널리 활용하며, 자연과학과 공학에도 활용됨.
Textbook
Introduction to Linear Algebray by Gilbert Strang
Contents
1 Vectors and Matrices
1.1 Vectors and Linear Combinations
1.2 Lengths and Angles from Dot Products
1.3 Matrices and Their Column Spaces
1.4 Matrix Multiplication AB and CR
2 Solving Linear Equations Ax = b
2.1 Elimination and Back Substitution
2.2 Elimination Matrices and Inverse Matrices
2.3 Matrix Computations and A = LU
2.4 Permutations and Transposes
3 The Four Fundamental Subspace
3.1 Vector Spaces and Subspaces
3.2 Computing the Nullspace by Elimination:A=CR
3.3 The Complete Solution to Ax = b
3.4 Independence, Basis, and Dimension
3.5 Dimensions of the Four Subspaces
4 Orthogonality
4.1 Orthogonality of Vectors and Subspaces
4.2 Projections onto Lines and Subspaces
4.3 Least Squares Approximations
4.4 Orthonormal Bases and Gram-Schmidt
4.5 The Pseudoinverse of a Matrix
5 Determinants
5.1 3 by 3 Determinants and Cofactors
5.2 Computing and Using Determinants
5.3 Areas and Volumes by Determinants
6 Eigenvalues and Eigenvectors
6.1 Introduction to Eigenvalues : Ax = λx
6.2 Diagonalizing a Matrix
6.3 Symmetric Positive Definite Matrices
6.4 Complex Numbers and Vectors and Matrices
6.5 Solving Linear Differential Equations
7 The Singular Value Decomposition (SVD)
7.1 Singular Values and Singular Vectors
7.2 Image Processing by Linear Algebra
7.3 Principal Component Analysis (PCA by the SVD)
8 Linear Transformations
8.1 The Idea of a Linear Transformation
8.2 The Matrix of a Linear Transformation
8.3 The Search for a Good Basis
9 Linear Algebra in Optimization
9.1 Minimizing a Multivariable Function
9.2 Backpropagation and Stochastic Gradient Descent
9.3 Constraints, Lagrange Multipliers, Minimum Norms
9.4 Linear Programming, Game Theory, and Duality
10 Learning from Data
10.1 Piecewise Linear Learning Functions
10.2 Creating and Experimenting
10.3 Mean, Variance, and Covariance
Reference
[1] Introduction to Linear Algebra, Sixth Edition (2023) by Gilbert Strang (gilstrang@gmail.com), ISBN : 978-17331466-7-8
