# 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 squares problem 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.