Schwerdtfeger H. Introduction to Linear Algebra and the Theory of Matrices 1961

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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