Linear algebra : concepts and methods

✍ Scribed by Martin Anthony; Michele Harvey

• Publisher: Cambridge University Press
• Year: 2012
• Tongue: English
• Leaves: 532
• Edition: 1
• Category: Library

✦ Synopsis

Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarize the material to help students consolidate their learning as they progress through the book

✦ Table of Contents

Content: Preliminaries : before we begin --
Sets and set notation --
Numbers --
Mathematical terminology --
Basic algebra --
Trigonometry --
A little bit of logic --
1. Matrices and vectors --
what is a matrix? --
Matrix addition and scalar multiplication --
Matrix multiplication --
Matrix algebra --
Matrix inverses --
Powers of a matrix --
The transpose and symmetric matrices --
Vectors --
Developing geometric insight --
Lines --
Planes --
Lines and hyperplanes --
2. Systems of linear equations --
Row operations --
Gaussian elimination --
Homogeneous systems and null space --
3. Matrix inversion and determinants --
Matrix inverse using row operations --
Determinants --
Matrix inverse using cofactors --
Leontief input-output analysis --
4. Rank, range and linear equations --
Rank of a matrix --
Rank and systems of linear equations --
Range --
5. Vector spaces --
Subspaces --
Linear span --
6. Linear independence, bases and dimension --
Linear independence --
Bases --
Coordinates --
dimension --
Basis and dimension --
7. Linear transformations and change of basis --
Range and null space --
Coordinate change --
Change of basis and similarity --
8. Diagonalisation --
Eigenvalues and eigenvectors --
Diagonalisation of a square matrix --
9. Applications of diagonalisation --
Powers of matrices --
Systems of difference equations --
Linear systems of differential equations --
10. Inner products and orthogonality --
Orthogonal matrices --
Gram-Schmidt orthonormalisation process --
11. Orthogonal diagonalisation and its applications --
Orthogonal diagonalisation of symmetric matrices --
Quadratic forms --
12. Direct sums and projections --
Direct sum of two subspaces --Orthogonal complements --
Projections --
Characterising projections and orthogonal projections --
Orthogonal projection onto the range of a matrix --
Minimising the distance to a subspace --
Fitting function to data : least squares approximation --
13. Complex matrices and vector spaces --
Complex numbers --
Complex vector spaces --
Complex matrices --
Complex inner product spaces --
Hermitian conjugates --
Unitary diagonalisation and normal matrices --
Spectral decomposition.
Abstract:

This thorough, yet concise, treatment of all the key topics also gives students a sound understanding of the underlying theory. Read more...