Mathematics II Chapters

System of linear equations; Row reduction and echelon forms; Vector equations; The matrix equation Ax = b; Applications of linear systems; Linear independence.

Introduction to linear transformations; The matrix representation of a linear transformation; Linear models in business, science, and engineering.

Matrix operations; Inverse of a matrix; Characterizations of invertible matrices; Partitioned matrices; Matrix factorization; Leontief input-output model; Subspaces of matrices.

Introduction to determinants; Properties of determinants; Cramer’s rule; Determinants in relation to volume and linear transformations.

Vector spaces and subspaces; Null spaces and column spaces; Linear transformations; Linearly independent sets; Bases; Coordinate systems.

Dimension and rank of vector spaces; Change of basis; Applications to difference equations; Applications to Markov Chains.

Eigenvectors and eigenvalues; Characteristic equation; Diagonalization; Linear transformations and eigenvectors; Complex eigenvalues; Discrete dynamical systems; Applications.

Inner product, length, and orthogonality; Orthogonal sets; Orthogonal projections; Gram-Schmidt process; Least squares problems; Applications to linear models; Inner product spaces.

Binary operations; Groups; Subgroups; Cyclic groups.

Rings and fields; Integral domains.