Invitation to Linear Algebra

Matrices and Linear Systems. Introduction to Matrices. Matrix Multiplication. Additional Topics in Matrix Algebra. Introduction to Linear Systems. The Inverse of a Matrix. Determinants. Introduction to Determinants. Properties of Determinants. Applications of Determinants. A First Look at Vector Spaces. Introduction to Vector Spaces. Subspaces of Vector Spaces. Linear Dependence and Independence. Basis and Dimension. The Rank of a Matrix. Linear Systems Revisited. More About Vector Spaces. Sums and Direct Sums of Subspaces. Quotient Spaces. Change of Basis. Euclidean Spaces. Orthonormal Bases. Linear Transformations. Introduction to Linear Transformations. Isomorphisms of Vector Spaces. The Kernel and Range of a Linear Transformation. Matrices of Linear Transformations. Similar Matrices. Matrix Diagonalization. Eigenvalues and Eigenvectors. Diagonalization of Square Matrices. Diagonalization of Symmetric Matrices. Complex Vector Spaces. Complex Vector Spaces. Unitary and Hermitian Matrices. Advanced Topics. Powers of Matrices. Functions of a Square Matrix. Matrix Power Series. Minimal Polynomials. Direct Sum Decompositions. Jordan Canonical Form. Applications. Systems of First Order Differential Equations. Stability Analysis of First Order Systems. Coupled Oscillations. Appendix. Solutions and Hints to Selected Exercises.