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Vectors, pure and applied : a general introduction to linear algebra /
"Many books in linear algebra focus purely on getting students through exams, but this text explains both the how and the why of linear algebra and enables students to begin thinking like mathematicians. The author demonstrates how different topics (geometry, abstract algebra, numerical analysi...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2013.
|
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Part I. Familiar vector spaces
- 1. Gaussian elimination
- Two hundred years of algebra
- Computational matters
- Detached coefficients
- Another fifty years
- 2. A little geometry
- Geometric vectors
- Higher dimensions
- Euclidean distance
- Geometry, plane and solid
- 3. The algebra of square matrices
- The summation convention
- Multiplying matrices
- More algebra for square matrices
- Decomposition into elementary matrices
- Calculating the inverse
- 4. The secret life of determinants
- The area of a parallelogram
- Rescaling
- 3 x 3 determinants
- Determinants of n × n matrices
- Calculating determinants
- 5. Abstract vector spaces
- The space Cn
- Abstract vector spaces
- Linear maps
- Dimension
- Image and kernel
- Secret sharing
- 6. Linear maps from Fn to itself
- Linear maps, bases and matrices
- Eigenvectors and eigenvalues
- Diagonalisation and eigenvectors
- Linear maps from C2to itself
- Diagonalising square matrices
- Iteration's artful aid
- LU factorisation
- 7. Distance preserving linear maps
- Orthonormal bases
- Orthogonal maps and matrices
- Rotations and reflections in R2and R3
- Reflections in Rn
- QR factorisation
- 8. Diagonalisation for orthonormal bases
- Symmetric maps
- Eigenvectors for symmetric linear maps
- Stationary points
- Complex inner product
- 9. Cartesian tensors
- Physical vectors
- General Cartesian tensors
- More examples
- The vector product
- 10. More on tensors
- Some tensorial theorems
- A (very) little mechanics
- Left-hand, right-hand
- General tensors
- Part II. General vector spaces
- 11. Spaces of linear maps
- A look at L(U, V)
- A look at L(U, U)
- Duals (almost) without using bases
- Duals using bases
- 12. Polynomials in L(U, U)
- Direct sums
- The Cayley-Hamilton theorem
- Minimal polynomials
- The Jordan normal form
- Applications
- 13. Vector spaces without distances
- A little philosophy
- Vector spaces over fields
- Error correcting codes
- 14. Vector spaces with distances
- Orthogonal polynomials
- Inner products and dual spaces
- Complex inner product spaces
- 15. More distances
- Distance on L(U, U)
- Inner products and triangularisation
- The spectral radius
- Normal maps
- 16. Quadratic forms and their relatives
- Bilinear forms
- Rank and signature
- Positive definiteness
- Antisymmetric bilinear forms
- Further exercises.