Algebra

1. Matrix Algebra: Definition of Matrix and different types of matrices with examples, Operations like addition, multiplication, transpose on matrices, properties of matrix algebra.

2. Determinant: Notion of permutations, sign of permutations, determinant for square matrices, properties of determinant, inverse of matrix.

3. System of linear equations: Homogeneous and non-homogeneous system of linear equations, Solution of system of linear equations by Matrix inversion method and Cramer's rule method.

4. Elementary row operations, Row Echelon Form, Reduced Row Echelon Form, Solution of system of linear equations by elementary row operations, Rank of a matrix, Criterion for a system of linear equation with no solution, with unique solution, and with infinitely many solutions.

5. (Abstract algebra) Group, Ring and Field: Definitions, Examples and basic properties.

6. Vector Space: Definition of vector space, subspace, and basic properties, Linear combination of a set of vectors, linearly dependent and independent set of vectors, Span of a set of vectors, Basis of a vector space, dimension of a vector space.

7. Linear transformation: Definition and examples of linear transformation, properties of linear transformations, Matrix representation of linear transformation.

8. (Classical algebra): Criteria for determining real positive, negative or complex roots of a polynomial, Descart's rule of sign, Synthesis method to find solutions.

9. Colloquium

M. Thulasidas, "Linear algebra for computer science"

J. Hefferon "Linear algebra"

P. A. Grillet "Abstract algebra"