Systems of linear equations, Matrices, Elementary row operations, Row-reduced echelon matrices .Vector spaces,
Subspaces, Bases and dimension, Ordered bases and coordinates.
Linear transformations, Rank-nullity theorem, Algebra of linear transformations, Isomorphism, Matrix representation, Linear
functionals, Annihilator, Double dual, Transpose of a linear transformation. Characteristic values and characteristic vectors of linear
transformations, Diagonalizability, Minimal polynomial of a linear transformation, Cayley-Hamilton theorem, Invariant subspaces,
Direct-sum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators,
Cyclic decomposition, Rational, Jordan forms. Inner product spaces, Orthonormal bases, Gram-Schmidt process.