Elements of Linear Algebra

✍ Scribed by Jason R. Wilson

• Publisher: VT Publishing
• Year: 2019
• Tongue: English
• Leaves: 145
• Category: Library

✦ Table of Contents

Part I — Vector Spaces
1 Introduction to Vector Spaces
1.1 Matrices and Vectors
1.2 Linear Systems
1.3 Vector Space Definition
1.4 General Results
1.5 Subspaces
1.6 Null Space of a Matrix
2 Span and Linear Independence
2.1 Linear Combinations and Span
2.2 Column Space of a Matrix
2.3 Linearly Independent Sets
2.4 Polynomial Spaces
2.5 Fundamental Theorem of Span and Independence
3 Basis and Dimension
3.1 Basis Definition
3.2 Finite-Dimensional Vector Spaces
3.3 Reduction and Extension
3.4 Subspace Dimension
3.5 Lagrange Polynomials
Part II — Linear Maps
4 Introduction to Linear Maps
4.1 Linear Map Definition
4.2 Null Space and Range
4.3 The Rank-Nullity Theorem
4.4 Composition and Inverses
5 Matrix Representations
5.1 Coordinate Maps
5.2 Matrix Representation Definition
5.3 Calculating Inverses
5.4 Change of Coordinates
6 Diagonalizable Operators
6.1 Diagonalizablity Definition
6.2 Eigenvalues and Eigenvectors
6.3 Determinants
6.4 Eigenspaces
6.5 Diagonalizability Test
Part III — Inner Product Spaces
7 Introduction to Inner Product Spaces
7.1 Inner Product Definition
7.2 Norms and Orthogonality
7.3 Orthogonal and Orthonormal Bases
7.4 Orthogonal Complements
8 Projections and Least Squares
8.1 Orthogonal Projections
8.2 The Gram-Schmidt Process
8.3 Best Approximation
8.4 Least Squares
9 Spectral Theorem and Applications
9.1 Orthogonal Diagonalization
9.2 The Spectral Theorem
9.3 Singular Value Decomposition
9.4 The Pseudoinverse