| LEC # | Topics | READINGS | PRACTICE PROBLEMS |
|---|---|---|---|
| 1 | Vector space, linear independence, dimension of a vector space, basis for vector space, components of a vector | pp. 1-8 | Problems 1.1-1.11 |
| 2 | Scalar product, length of vector, distance between vectors, angle between vectors, orthonormal basis | pp. 9-17 | Problems 1.12-1.20 |
| 3 | Linear transformations, invariant subspace, eigenvalue problem | pp. 18-20 and 23-26 | Problems 2.1, 2.3, 2.6, and 2.17 (except questions about singular/non-singular/inverse transforms) |
| 4 | Null space, singular/non-singular linear transformations, inverse, components of a linear transformation | Chapter 2 | Problems 2.1-2.5, 2.8, 2.9, 2.11, and 2.15-2.17 |
| 5 | Components of a linear transformation, components in different bases, scalar invariants, cartesian tensors, symmetric tensors, skew-symmetric tensors | pp. 27-32 and 42-46 | Problems 3.1-3.12 |
| 6 | Eigenvalues of a symmetric tensor, principal basis, positive-definite tensor, orthogonal tensor, proper/improper, orthogonal tensor | pp. 42-52 (except tensor products) and 56-57 | Problems 3.13-3.18, 3.20, and 3.24-3.26 |
| 7 | Tensor product of 2 vectors, polar decomposition of a non-singular tensor | pp. 44 and 57-59 Also read chapters 2 and 3 in Abeyaratne, Rohan. Lecture Notes on the Mechanics of Elastic Solids. | Problems 3.3-3.7, 3.11-3.16, 3.22, and 3.23 |
Courses:
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations >> Content Detail
Table of Contents
Study Materials
Readings
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Texts
Most of the readings and all practice problems are from the course text:
Knowles, J. K. Linear Vector Spaces and Cartesian Tensors. New York, NY: Oxford University Press, 1998. ISBN: 9780195112542.
Abeyaratne, Rohan. Lecture Notes on the Mechanics of Elastic Solids. Vol. 1, A Brief Review of Some Mathematical Preliminaries. Free e-book, 2006. (PDF - 1.4 MB)#
Further References
Gel'fand, I. M. Lectures on Linear Algebra. New York, NY: Dover, 1989. ISBN: 9780486660820.
Halmos, P. R. Finite Dimensional Vector Spaces. Princeton, NJ: Van Nostrand-Reinhold, 1958. ISBN: 9780387900933.
Gel'fand, I. M., and S. V. Fomin. Calculus of Variations. Englewood Cliffs, NJ: Prentice Hall, 1963.
Giaquinta, M., and S. Hilderbrandt. Calculus of Variations I. New York, NY: Springer, 1996. ISBN: 9780387506258.
Troutman, J. L. Variational Calculus with Elementary Convexity. New York, NY: Springer-Verlag, 1983. ISBN: 9780387907710.