Two of the biggest 'branches' of mathematics are
analysis and algebra. Essentially the first course
in 'analysis' is calculus.

As in Simmons (in this list), the two most important
pillars of analysis are continuity and linearity.

The first place to learn linearity well is 'linear
algebra' -- or, much the same thing, matrix theory
or finite dimensional vector spaces.

Then the algebraic aspects can be emphasized and can
lead to applications in, say, the theory of error
correcting codes.

For the analysis aspects, can continue with Hilbert
and Banach spaces and more and also various
applications.

Much of the mathematics in applied mathematics,
science, engineering, and technology is linear
algebra.

Here is a list of books relevant to linear algebra
and/or to the course I took from Roger Horn. This
list also contains maybe 60% of what I'd worked from
before Horn's course that let me do well in that
course.

========================================

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.