Definition and Examples of Vector Spaces, Subspaces, Algebra of subspaces, Linear span, Linear dependence and independence of vectors, Basis and Dimension of a vector space, Basis and dimension of subspace, Direct sums and complements.
Linear Transformations, Rank and Nullity of a linear transformation, Vector space of linear transformations.
Linear transformation and matrices, Change of basis.
Characteristic roots and characteristic vectors, Algebraic and Geometric multiplicity of a characteristic value, Cayley‑Hamilton theorem, Diagonalizable operators and matrices. Minimal polynomial of a linear operator (matrix).