Linear Algebra and Applications

1.1 Introduction to Systems of Linear Equations +

1.2 Gaussian Elimination +

1.3 Matrices and Matrix Operations +

1.4 Inverses; Algebraic Properties of Matrices +

1.5 Elementary Matrices and a Method for Finding A⁻¹ +

1.6 More on Linear Systems and Invertible Matrices +

1.7 Diagonal, Triangular, and Symmetric Matrices +

1.8 Introduction to Linear Transformations +

1.9 Compositions of Matrix Transformations +

1.10 Applications of Linear Systems +

2.1 Determinants by Cofactor Expansion +

2.2 Evaluating Determinants by Row Reduction +

2.3 Properties of Determinants; Cramer’s Rule +

3.1 Vectors in 2-Space, 3-Space, and n-Space +

3.2 Norm, Dot Product, and Distance in ℝⁿ +

3.3 Orthogonality +

4.1 Real Vector Spaces +

4.2 Subspaces +

4.3 Spanning Sets +

4.4 Linear Independence +

4.5 Coordinates and Basis +

4.6 Dimension +

4.7 Change of Basis +

4.8 Row Space, Column Space, and Null Space +

4.9 Rank, Nullity, and the Fundamental Matrix Spaces +

5.1 Eigenvalues and Eigenvectors +

5.2 Diagonalization +

5.4 Differential Equations +

6.1 Inner Products +

6.2 Angle and Orthogonality in Inner Product Spaces +

6.3 Gram–Schmidt Process; QR-Decomposition +

7.1 Orthogonal Matrices +

7.2 Orthogonal Diagonalization +

8.1 General Linear Transformations +

8.2 Compositions and Inverse Transformations +

8.4 Matrices for General Linear Transformations +