Elements of Linear Algebra

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