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Linear Algebra

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Author(s): S. J. Wadsley, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part IB Lecture Notes
Publisher: University of Cambridge
Year: 2015

Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; linear algebra; matrices; matrix; vectors; vector

Introduction
Vector spaces
Definitions and examples
Linear independence, bases and the Steinitz exchange lemma
Direct sums
Linear maps
Definitions and examples
Linear maps and matrices
The first isomorphism theorem and the rank-nullity theorem
Change of basis
Elementary matrix operations
Duality
Dual space
Dual maps
Bilinear forms I
Determinants of matrices
Endomorphisms
Invariants
The minimal polynomial
Aside on polynomials
Minimal polynomial
The Cayley-Hamilton theorem
Multiplicities of eigenvalues and Jordan normal form
Bilinear forms II
Symmetric bilinear forms and quadratic forms
Hermitian form
Inner product spaces
Definitions and basic properties
Gram-Schmidt orthogonalization
Adjoints, orthogonal and unitary maps
Spectral theory