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.