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Preface Introduction Geometrical Foundations - Systems of Linear Equations The Number Space Linear Dependence of Points in Number Space The Replacement Procedure. The Theorem of Grassmann and Steinitz Linear Equations - Rank of a Matrix A Matrix composed of Rows — Transposition Geometrical Interpretation. Parallel Manifolds Determinants Determinants and the Solution of Linear Systems Determinants and the Rank of Matrices Linear Homogeneous Transformations. Rational Operations on Matrices Substitutions and Transformations Linear Homogeneous Transformations and their Matrices The Formal Laws of Matrix Algebr Regular and Singular Matrices Fields. Complex and hypercomplex Numbers Equivalence, Congruence, Bilinear and Quadratic Forms Elementary Transformations of a Matrix Equivalence Bilinear and Quadratic Forms. Congruence Symmetric Matrices with Complex Elements Real Symmetric Matrices. Law of Inertia Hermitean Matrices and Hermitean Forms Skew-symmetric Matrices Groups of Matrices. Similarity The Notion of Group in Linear Algebra Similarity. The Characteristic Polynomial Similarity of Groups. Congruence with respect to a Group. Group Representations The Orthogonal Group. Reflections The Unitary Group. - Normal Matrices The Eigen Value Problem of a Normal Matrix Orthogonal and Skew-symmetric Matrices The Symplectic Groups Projective Theory of Null Systems Vector Spaces over a Field Cayley-Hamilton's Identity Summary of the Principal Problems in Chapters III and IV Index