Elementary Linear Algebra: A Matrix Approach by Stephen Spence, Lawrence Instel, and Arnold Friedberg, 2nd Edition.
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Chapter 1 Matrices, Vectors, and Systems of Linear Equations
1.1 Matrices and Vectors
1.2 Linear Combinations, Matrix-Vector Products, and Special Matrices
1.3 System of Linear Equations
1.4 Gaussian Elimination
1.6 The Span of a Set of Vectors
1.7 Linear Dependence and Linear Independence
Chapter 2 Matrices and Linear Transformations
2.1 Matrix Multiplication
2.3 Invertibility and Elementary Matrices
2.4 The Inverse of a Matrix
2.6 The LU Decomposition of a Matrix
2.7 Linear Transformations and Matrices
2.8 Composition and Invertibility of Linear Transformations
Chapter 3 Determinants
3.1 Cofactor Expansion
3.2 Properties of Determinants
Chapter 4 Subspaces and Their Properties
4.1 Subspaces
4.2 Basis and Dimension
4.3 The Dimension of Subspaces Associated with a Matrix
4.4 Coordinate Systems
4.5 Matrix Representations of Linear Operators
Chapter 5 Eigenvalues, Eigenvectors, and Diagonalization
5.1 Eigenvalues and Eigenvectors
5.2 The Characteristic Polynomial
5.3 Diagonalization of Matrices
Additional Topic: Cayley--Hamilton Theorem for Matrices
Chapter 6 Orthogonality
6.7 Singular Value Decomposition