What's the point of duality? I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of duality?
 A: Duality is a simple way to make new vector spaces.  These dual spaces are useful in functional analysis, for example when you want to define the integral of a function, or you want to analyze a probability distribution.  In this case there's a vector space of functions and a linear way to map those functions to numbers, which is natural to describe as an element of the dual space.
For finite-dimensional vector spaces, the dual is not so interesting because it looks like the original vector space.  So it may not be very exciting in a standard undergraduate course.  But for infinite dimensions, things are more interesting.
I have just been refreshing my memory from the Wikipedia article on Dual Space, which is a good summary.
A: For what it's worth, you're not the only one having trouble seeing the immediate relevance of dual spaces. In the preface to Michael Artin's algebra textbook, he says:

(2) The book is not intended for a "service course," so technical points should be presented only if they are needled in the book.
(3) All topics discussed should he important for the average mathematician.
[...] Sometimes the exercise of deferring material showed that it could be deferred forever -- that it was not essential. This happened with dual spaces and multilinear algebra, for example, which wound up on the floor as a consequence of the second principle.

When I read that as an undergraduate I thought "yeah, whatever" -- since what else could I do, not knowing what it was I was missing.
However, later when I came to differential geometry and tensor calculus (which I needed for general relativity) it turned out that duality is absolutely essential there. Then it wasn't very satisfying to lack the general algebraic grounding to fully appreciate what was happening. The books I were using did provide the bare essentials I needed to follow along, but it was also clear that there was a nice algebraic systematic hiding underneath all that which I didn't get to see all of. And it would certainly have been helpful to know that general theory before embarking on differential geometry.
A: This will probably not be apparent in a linear algebra course, but duality is the workhorse of optimization. Roughly speaking, you can often frame an optimization problem as trying to minimize some quantity subject to linear constraints (that form a matrix). Then in order to solve this problem you usually need to understand the "dual" problem, which is another optimization problem whose constraint matrix is the dual to the original constraint matrix. It is by understanding the primal and dual spaces simultaneously that you can prove that you have an optimal (or near optimal) solution for your problem. 
A: Expanding on Hew Wolff's comment: in algebraic topology, a major theme is functoriality, which roughly boils down to the idea that when we associate algebraic objects to topological spaces, we should similarly obtain any maps between the algebraic objects from maps between topological spaces so that the algebra truly reflects the topology.
Homology attaches a graded sequence of (additive) abelian groups $H_*(X)$ to a space $X$, and associates a map between such groups $f_*:H_*(X)\to H_*(Y)$ to a continuous map $f:X\to Y$. If $X$ is an $n$-dimensional manifold, the sequence $H_*(X)$ contains a (potentially) nonzero graded piece $H_i(X)$ for each $i=0,...,n$ which tells us, roughly, the number of $i$-dimensional holes in $X$. If we appeal to some intuition from linear algebra, where we can  multiply together two vector spaces of dimensions $m$ and $n$ to obtain a vector space of dimension $m+n$, this idea of functoriality motivates us to ask if we can somehow multiply lower-dimensional information in homology to obtain higher-dimensional information.
To obtain such a product map $H_m(X)\times H_n(X)\to H_{n+m}(X)$, we would need a continuous map $X\times X\to X$; unfortunately, the only halfway decent choices we have a priori are the projection maps onto factors, which obviously throw away a good deal of information. On the other hand, the dual construction of cohomology has a product map $H^m(X)\times H^n(X)\to H^{m+n}(X)$ induced by the diagonal embedding $X\to X\times X$ which sends $x\mapsto (x,x)$.
Mike, hopefully this is somewhat understandable. I realize that it goes somewhat far afield, but that is somewhat necessary to answer the question. The main thing to understand is that your confusion is due to the artifice of presentation; duality arose in nature before it was formalized; it is really quite ubiquitous. Also, I should note that most of this discussion sticks quite close to Allen Hatcher's book.
