Linear Algebra

Linear algebra is a branch of mathematics concerning linear equations and linear maps, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics.

In a course on linear algebra, one may deal with the following topics:

  • Systems of linear equations

  • Row reductions and echelon forms

  • Vector and matrix equations

  • Solution sets of linear systems

  • Linear independence

  • Linear transformations and their matrices

  • Matrix operations, the inverse of a matrix, and characterizations of invertible matrices

  • Matrix factorization (LU factorization)

  • Subspaces of n-dimensional real vector spaces

  • Introduction to determinants and their properties

  • Cramer's rule and volume, and linear transformations

  • Null and column spaces

  • Bases and the dimension of a real vector space

  • Rank

  • Change of basis

  • Eigenvectors, eigenvalues, and eigenspaces

  • The characteristic equation

  • Matrix diagonalization

  • Inner product, length, and orthogonality

  • Orthogonal sets and projections

  • The Gram-Schmidt process

  • The Least square problems and linear regression

  • Diagonalization of symmetric matrices

  • Quadratic forms

  • Constrained optimization

  • The singular value decomposition (SVD)

For more details, refer to the following book:

Lay, D. C. (2016). Linear Algebra and its applications 5th edition. Pearson.

Ordinary Differential Equations

Semiring Theory

Abstract Algebra

Commutative Algebra