### 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. *A Brief Review of Some Mathematical Preliminaries*. Volume 1 in *Lecture Notes on the Mechanics of Elastic Solids*. [Free e-book.]

### 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.

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. |
Problems 3.3-3.7, 3.11-3.16, 3.22, and 3.23 |