The Determinant'S Properties and Calculation. the Adjoint Matrix. Direct Sums and Projections. Invariant Subspaces, Quotient Spaces. the Polynomial Ring: Roots and Factorization, Euclidean Algorithm. the Minimal and Characteristic Polynomials and the Cayley Hamilton Theorem. Existence and Uniqueness of the Jordan Form. Inner Product and the Cauchy Shwartz Inequality. Orthogonal Bases and the Gramm-Schmidt Algorithm. Duality, Riesz Theorem, the Adjoint Operator. Selfadjoint, Unitary and Normal Operators. Diagonalization and the Spectral Decomposition. the Polar Decomposition. Bilinear and Quadratic Forms.