Elements of Linear and Multilinear Algebra

This set of notes is an activity-oriented introduction to linear and multilinear algebra. The great majority of the most elementary results in these subjects are straightforward and can be verified by the thoughtful student. Indeed, that is the main point of these notes — to convince the beginner that the subject is accessible. In the material that follows there are numerous indicators that suggest activity on the part of the reader: words such as 'proposition', 'example', 'theorem', 'exercise', and 'corollary', if not followed by a proof (and proofs here are very rare) or a reference to a proof, are invitations to verify the assertions made.These notes are intended to accompany an (academic) year-long course at the advanced undergraduate or beginning graduate level. (With judicious pruning most of the material can be covered in a two-term sequence.) The text is also suitable for a lecture-style class, the instructor proving some of the results while leaving others as exercises for the students.This book has tried to keep the facts about vector spaces and those about inner product spaces separate. Many beginning linear algebra texts conflate the material on these two vastly different subjects.Contents: Vector SpacesLinear TransformationsThe Language of CategoriesThe Spectral Theorem for Vector SpacesThe Spectral Theorem for Inner Product SpacesA Brief Review of Differential CalculusMultilinear Maps and DeterminantsTensor AlgebrasDifferential ManifoldsDifferential Forms on ManifoldsHomology and CohomologyStokes' TheoremGeometric AlgebraClifford Algebras
Readership: Upper division undergraduates, beginning graduate students, instructors of linear and multilinear algebra.Linear Algebra;Algebra;Vectors;Vector Spaces;Subspaces;Dependence;Span;Basis;Linear;Transformation;Kernel;Null Space;Eigenvalue;Eigenvector;Spectrum;Spectral Theorem;Matrix;Inner Product;Orthogonal;Perpendicular;Cauchy-Schwarz;Norm;Adjoint;Self-Adjoint;Hermitian;Diagonalization;Dimension;Real;Complex;Geometric Algebra;Multilinear;Projection;Tensors;Span;Rank;Nullity;Similarity;Isomorphism;Annihilator;Decomposition;Bilinear;Convexity;Clifford Algebras;Differential Forms;Differential Manifolds;Grassmann Algebras;Homology;Cohomology;De Rham Cohomology;Wedge Product;Cotangent Bundle;Commutative Diagrams;Exterior Product;Functors;Coboundary;Cochain;Cocycle;Quotients;Riesz-Fréchet Theorem;Simplex;Skew-Symmetric0Key Features:Provides an activity-oriented introduction to linear and multilinear algebraMakes a careful distinction between the facts true of vector spaces and those about inner product spacesIncludes an in-depth discussion of differentiable forms as a significant application of multilinear algebraIncludes brief introductions to geometric algebra and Clifford algebrasIs suitable as a basis for a lecture course