Expalin what group theory is about through the example of the Eucledia group
http://en.wikipedia.org/wiki/Euclidean_group
Recommended books from greycat @ https://news.ycombinator.com/item?id=6882107
Linear Algebra
Horn's course was close to Roger A. Horn, Charles R. Johnson, 'Matrix Analysis', 0-521-38632-2, Cambridge University Press, 1990. with also a few topics from Roger A. Horn, Charles R. Johnson, 'Topics in Matrix Analysis', 0-521-46713-6, Cambridge University Press, 1994.
The course also used for reference and some topics and exercises Richard Bellman, 'Introduction to Matrix Analysis: Second Edition', McGraw-Hill, New York, 1970. This book is just packed with little results; at some point can get the impression that the author went on and on … writing. Bellman was a very bright guy in mathematics, engineering, and medicine.
Relatively easy to read and relatively close to applications, and another book Horn's course used as a reference, is Ben Noble, 'Applied Linear Algebra', Prentice-Hall, Englewood Cliffs, NJ, 1969. Some edition of this book may be a good place to start for a student interested in applications now.
About the easiest reading in this list, and my first text on linear algebra, was D. C. Murdoch, 'Linear Algebra for Undergraduates', John Wiley and Sons, New York, 1957. This book is awfully old, but what it has are still the basics.
For my undergraduate honors paper I made some use of, and later read carefully nearly all of, Evar D. Nering, 'Linear Algebra and Matrix Theory', John Wiley and Sons, New York, 1964. The main part of this book is a relatively solid start, maybe a bit terse and advanced for a first text. The book also has in the back a collection of advanced topics, some of which might be quite good to know at some point and difficult to get elsewhere. One of the topics in the back is linear programming, and for that I'd recommend something else, e.g., Chv'atal and/or Bazaraa and Jarvis in this list.
Likely the crown jewel of books on linear algebra is Paul R. Halmos, 'Finite-Dimensional Vector Spaces, Second Edition', D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. Halmos wrote this in about 1942 when he was an assistant to von Neumann at the Institute for Advanced Study. The book is intended to be a finite dimensional introduction to Hilbert space theory, or how to do linear algebra using mostly only what also works in Hilbert space. It's likely fair to credit von Neumann with Hilbert space. The book is elegant. Apparently at one time Harvard's course Math 55, with a colorful description at, http://www.american.com/archive/2008/march-april-magazine-co… used this text by Halmos and also, as also in this list, Rudin's 'Principles' and Spivak.
Long highly regarded as a linear algebra text is Hoffman and Kunze, 'Linear Algebra, Second Edition', Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
Numerical Methods
If want to take numerical computations in linear algebra seriously, then consider the next book or something better if can find it George E. Forsythe and Cleve B. Moler, 'Computer Solution of Linear Algebraic Systems', Prentice-Hall, Englewood Cliffs, 1967.
Multivariate Statistics My main start with multivariate statistics was N. R. Draper and H. Smith, 'Applied Regression Analysis', John Wiley and Sons, New York, 1968. Apparently later editions of this book remain of interest.
A relatively serious book on 'regression analysis' is C. Radhakrishna Rao, 'Linear Statistical Inference and Its Applications: Second Edition', ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967.
Three famous, general books on multivariate statistics are Maurice M. Tatsuoka, 'Multivariate Analysis: Techniques for Educational and Psychological Research', John Wiley and Sons, 1971. William W. Cooley and Paul R. Lohnes, 'Multivariate Data Analysis', John Wiley and Sons, New York, 1971. Donald F. Morrison, 'Multivariate Statistical Methods: Second Edition', ISBN 0-07-043186-8, McGraw-Hill, New York, 1976.
Analysis of Variance A highly regarded first book on analsis of variance and experimental design is George W. Snedecor and William G. Cochran, 'Statistical Methods, Sixth Edition', ISBN 0-8138-1560-6, The Iowa State University Press, Ames, Iowa, 1971. and a famous, more mathematical, book is Henry Scheff'e, 'Analysis of Variance', John Wiley and Sons, New York, 1967.
Linear Optimization A highly polished book on linear programming is Vav sek Chv'atal, 'Linear Programming', ISBN 0-7167-1587-2, W. H. Freeman, New York, 1983.
Nicely written and with more emphasis on the important special case of network flows is Mokhtar S. Bazaraa and John J. Jarvis, 'Linear Programming and Network Flows', ISBN 0-471-06015-1, John Wiley and Sons, New York, 1977.
A grand applied mathematics dessert buffet, based on Banach space and the Hahn-Banach theorem is David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969.
Mathematical Analysis Relevant to Understanding Linearity
Long the first place a math student gets a fully serious encounter with calculus and closely related topics has been Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York, 1964. The first chapters of this book do well as an introduction to metric spaces, and that work applies fully to vector spaces.
A nice place to get comfortable doing mathematics in several dimensions is Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts, 1965. Some of the material here is also good for optimization.
Another place to get comfortable doing mathematics in several dimensions is Michael Spivak, {\it Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,\/} W.\ A.\ Benjamin, New York, 1965.\ \
The first half, the 'real' half of the next book has polished introductions to Hilbert and Banach spaces which are some of the most important vector spaces Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York, 1966.
An elegant introduction to how to get comfortable in metric space is George F. Simmons, 'Introduction to Topology and Modern Analysis', McGraw Hill, New York, 1963.
Ordinary Differential Equations
Linear algebra is important, as some points crucial, for ordinary differential equations, a polished introduction from a world expert is Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961. replyparent