System of linear equations, Row reduction and Echelon forms, Vector equations, The matrix equations $Ax=b$, Applications of linear system, Linear independence.

Introduction to linear transformations, the matrix of a linear Transformation, Linear models in business, science, and engineering.

Matrix operations, The inverse of a matrix, Characterizations of invertible matrices, Partitioned matrices, Matrix factorization, The Leontief input output model, Subspace of $R^n$ Dimension and rank.

Introduction, Properties, Cramer's rule, Volume and linear transformations.

Vector spaces and subspaces, Null spaces, Column spaces, and Linear transformations, Linearly independent sets: Bases, Coordinate systems.

Dimension of vector space and Rank, Change of basis, Applications to difference equations, Applications to Markov Chains.

Eigenvectors and Eigenvalues, The characteristic equations, Diagonalization, Eigenvectors and linear transformations, Complex eigenvalues, Discrete dynamical systems, Applications to differential equations.

Inner product, Length, and orthogonality, Orthogonal sets, Orthogonal projections, The Gram-Schmidt process, Least squares problems, Application to linear models, Inner product spaces, Applications of inner product spaces.

Binary Operations, Groups, Subgroups, Cyclic Groups.

Rings and Fields, Integral domains.