Quotient space (linear algebra) In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N). The equivalence class of x is often denoted [x] = x + N since it is given by The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by α[x] = [αx] for all α ∈ K, and[x] + [y] = [x+y].
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map. Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. More generally, if V is an (internal) direct sum of subspaces U and W, Introduction to vectors and tensors, Vol 1: linear and multilinear algebra. Online Mathematics Textbooks. Professor Jim Herod and I have written Multivariable Calculus ,a book which we and a few others have used here at Georgia Tech for two years. We have also proposed that this be the first calculus course in the curriculum here, but that is another story....
Although it is still in print, Calculus,by Gilbert Strang is made available through MIT's OpenCourseWare electronic publishing initiative. Here is one that has also been used here at Georgia Tech. Linear Methods of Applied Mathematics, by Evans Harrell and James Herod. Yet another one produced at Georgia Tech is Linear Algebra, Infinite Dimensions, and Maple, by James Herod. I have also written a modest book, Complex Analysis, which I have used in our introductory undergraduate complex analysis course here. Banach-Tarski Paradox -- Math Fun Facts. Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original?
This theorem is known as the Banach-Tarski paradox. So why can't you do this in real life, say, with a block of gold? If matter were infinitely divisible (which it is not) then it might be possible. But the pieces involved are so "jagged" and exotic that they do not have a well-defined notion of volume, or measure, associated to them. In fact, what the Banach-Tarski paradox shows is that no matter how you try to define "volume" so that it corresponds with our usual definition for nice sets, there will always be "bad" sets for which it is impossible to define a "volume"!
Presentation Suggestions: Students will find this Fun Fact hard to believe. The Math Behind the Fact: First of all, if we didn't restrict ourselves to rigid motions, this paradox would be more believable. Layman's Guide to the Banach-Tarski Paradox. Preliminaries First of all, let's nail down what exactly we're talking about so that we're all on the same page. First and foremost, we're talking about a mathematical sphere, not a physical sphere, although I'd like to use an analogy with physical spheres to describe one possible way to intuit the Banach-Tarski Paradox. By mathematical sphere, I mean the set of points that lie within a 3-dimensional spherical area in ℜ3, where ℜ is the set of all real numbers.
For simplicity, let's assume a radius of 1, so our sphere would be the set: S = {(x,y,z) | x2+y2+z2 <= 1 } One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number2 of points. 2Or, to be precise, c points, where c is the cardinality of the continuum. Now let's move on to the paradox itself.
The Banach-Tarski Paradox (Before you dismiss this notion outright, let me state that mathematically infinite objects do not always behave intuitively. Drunken Walker and Fly -- Math Fun Facts. Imagine a drunken person wandering on the number line who starts at 0, and then moves left or right (+/-1) with probability 1/2. What is the probability that the walker will eventually return to her starting point Answer: probability 1.
What about a random walk in the plane, moving on the integer lattice points, with probability 1/4 in each of the coordinate directions? What's the chance of return to the starting point? Answer: also probability 1. OK, now what about a drunken fly, with 6 directions to move, probability 1/6? Presentation Suggestions: Try to give a little insight by illustrating a random walk on the line for several steps. The Math Behind the Fact: A probabilist would say that simple random walks on the line and plane are recurrent, meaning that with probability one the walker would return to his starting point, and that simple random walks in dimensions 3 and higher are transient, meaning there is a positive probability that he will never return!