Introductory Linear Algebra

1 Matrices
1.1 Preliminaries
1.2 Matrices
1.3 Properties of a Matrix
1.4 Block Matrices

2 System of Linear Equations
2.1 Gauss and Gauss-Jordan Elimination Method
2.2 Inverse Matrix
2.3 Elementary Matrix Multiplication
2.4 Column Operations

3 Vector Spaces
3.1 Vector Spaces
3.2 Geometry of Vectors and its Applications
3.3 Subspaces
3.4 Basis of a Vector Space
3.5 Row and Column Spaces

4 Linear Transgormations and Matrices
4.1 Linear Transformations
4.2 Matrix Representation of a Linear Transformation
4.3 Composition of Linear Transformations and Matrix Multiplications
4.4 Isomorphisms
4.5 Change of Bases
4.6 Dual Spaces

5 Determinants
5.1 Definition of Determinant
5.2 Properties of Determinant
5.3 Cramer's Rule
5.4 LU Decomposition and Determinant

6 Inner Product
6.1 Inner Product
6.2 Gram-Schmidt Theorem
6.3 Inner Product and Matrices
6.4 Lines and Planes

7 Eigenvalues and their Applications
7.1 Hamilton-Cayley Theorem
7.2 Eigenvalues and thier Applications
7.3 Diagonalization of a Square Matrix
7.4 Symmetric Matrices
7.5 Quadratic Forms

8 Joredan and Other Canonical Forms
8.1 Quotient Spaces
8.2 Minimal Polynomials and Linear Trasformations
8.3 Triangular Forms
8.4 Jordan Canonical Forms

Bibiliography