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Contents

• V Full version
• 0 Introduction
• 1 Vector spaces
• 1.1 Definitions and examples
• 1.2 Linear independence, bases and the Steinitz exchange lemma
• 1.3 Direct sums
• 2 Linear maps
• 2.1 Definitions and examples
• 2.2 Linear maps and matrices
• 2.3 The first isomorphism theorem and the rank-nullity theorem
• 2.4 Change of basis
• 2.5 Elementary matrix operations
• 3 Duality
• 3.1 Dual space
• 3.2 Dual maps
• 4 Bilinear forms I
• 5 Determinants of matrices
• 6 Endomorphisms
• 6.1 Invariants
• 6.2 The minimal polynomial
• 6.3 The Cayley-Hamilton theorem
• 6.4 Multiplicities of eigenvalues and Jordan normal form
• 7 Bilinear forms II
• 7.1 Symmetric bilinear forms and quadratic forms
• 7.2 Hermitian form
• 8 Inner product spaces
• 8.1 Definitions and basic properties
• 8.2 Gram-Schmidt orthogonalization
• 8.3 Adjoints, orthogonal and unitary maps
• 8.4 Spectral theory