linear algebra

What is linear algebra, anyway?

Most people who've taken or heard of the course will say, "Oh, that course where you do stuff with matrices."  For the uninitiated, you could imagine a matrix as a rectangular 2D grid of numbers with rows and columns, much like an excel spreadsheet, that represents various things like coefficients of algebra equations or

(Hilariously, according to Wikipedia, "matrix" is Latin for "womb" - a term introduced in 1848 by James Joseph Sylvester, an English mathematician who made early contributions to matrix algebra and invented many other mathematical terms such as "discriminant".  I instantly liked the guy because he liked to read and translate poetry, got paid in gold by Johns Hopkins University, and founded the American Journal of Mathematics.)

But linear algebra is a lot more general than that.  From my class notes, we could loosely call linear algebra the study of "stuff" that is flat and not curved: for example, lines or planes but not knots or parachutes.  As it turns out, you can add "stuff" or multiply it by numbers (scalars is the fancy term) and still preserve these properties.

PROPERTIES OF VECTOR SPACES
This is awfully vague and inexact, so let's try to say what kinds of "stuff" can be added and multiplied by scalars.  We can think of this "stuff" as a set of things - let's call them vectors, but you could also have the things in this set be functions, polynomials, matrices, or any other objects - with certain properties: If you add two of these vectors (call them v and w) together, you get a third vector (v+w) that still belongs to that set.  If you multiply vector v by a number a (more generally called a scalar), you get another vector av.  So let's call the set of things that can be added and multiplied by scalars pairwise a vector space.

What's so special about this set of vectors?  Well, it doesn't matter what order you add or multiply them in (commutivity of addition: u+v=v+u).  If you had three vectors, it doesn't matter whether you add the first and second, then add the third or add the second and third, then add the first (associativity: (u+v)+w=u+(v+w) ).  Ditto for multiplication. as a*(k*v)=k*(a*v), where k and a are scalars.  You can add zero to a vector and get the same vector (identity of addition: 0+v=v) and multiply by one and get the same vector (identity of multiplication: 1v=v).  Last but not least, you can add two vectors then multiply by a scalar or multiply both vectors by a scalar and add them and get the same answer (distributivity: k*(v+w)=k*v+k*w).

SCALARS
These "numbers" or scalars could be real or complex numbers, but they could also be more exotic things like quotient spaces, surreal numbers, finite fields, or p-adic numbers.   Don't ask me what those are, though... that's just a list from class because it sounds cool. :)


http://en.wikipedia.org/wiki/Linear_algebra
Lax, Peter  D. "Chapter 1: Fundamentals."  Linear Algebra and Its Applications, 2nd Ed.  Wiley-Interscience: Hoboken, NJ (2007).