## Sunday, August 15, 2010

### What are vectors?

As I've said before, science is social-- oops. Wrong mantra. What I meant to say is that vectors are an abstract way of describing a pattern. Specifically, the vectorspace axioms formally describe a kind of mathematical object, the vector, that encapsulates the geometric and algebraic properties of a large class of seemingly disparate objects. By using the vectorspace axioms, we will be able see that lists of numbers such as are vectors, as are arrows on the 2D plane.

Rather than describe how to do so myself, though, I will try something different. Vectors are important in much of physics, and so lots of people have already written much about them. Thus, for the bulk of the work in describing vectors, I will defer to these other writings. A very physics-oriented approach can be found over at Dot Physics, starting with a trig-based introduction to vectors, followed by a discussion of how to represent vectors. An alternate physics-motivated discussion of vectors can be found at HyperPhysics.

For the more mathematically motivated amongst us, Wikipedia has a good page describing a very special family of vector spaces called ℝn that is used to describe points in Euclidean space. MathWorld has a few good articles on vectors, including a technical definition and listing of properties and a more concise listing of the vectorspace axioms. Finally, the Unapologetic Mathematician derives vectorspaces from a more general construction called a module (warning: not for the feint of math).

To understand why we care about vectors in quantum information and computation, however, takes one more observation. A quantum state can be written as a linear combination of some set of basis states. For example, an arbitrary qubit state can be written as . This important property means that quantum states are a kind of vector in what we call a Hilbert space. This has some profound implications for how we think of and manipulate quantum states, as we shall explore in forthcoming posts.

Caleb wherry said...

I am looking forward to seeing where this goes. I need to jump on board and actually post science-y/math-y stuff as well. Now I just need to think of a starting topic...

It's only recently that I've felt anywhere near confident enough to delve into science here, so I'm happy to hear that it's received well. As for starting topics, I'm sure you'll think of something wonderful.

Luke said...

I looked through those introductions and one thing I don't see emphasised, which I would have loved to see emphasised, is how vectors are rank 1 tensors and thus entirely co-ordinate dependent. Otherwise, choosing a different co-ordinate system, the components change!

know for matrices, we always talk about change of basis and the best basis for doing a problem - the energy eigenbasis or computational basis or whatever people like to call it - yet I've never seen people talk about how to do so for vectors. I saw it arise maybe once or twice, but it was never dwelt upon. I know when I learnt GR I had to unlearn the whole way we learn vectors and relearn the better - in my opinion - way to do it. Heck the whole thing about \partial_{i} and dx^{i} forming a basis for our covariant and contravarient spaces makes sense, what with how they transform. Of course, I don't encourage that first year professor in physics telling students that. It'd be nice to see more introductions, specifically for physics, put more emphasis on vectors depend on the basis chosen.

I suppose for quantum mechanics it's not really needed but since we're always diagonalising the Hamiltonian and telling students that the matrix is really the same as the complicated one before, might as well emphasis that for vectors as well.

Luke said...

Yes, I was planning on tackling bases a bit, but I can pretty well guarantee that it's not going to be at the level of rigor that will satisfy the kind of argument you've made here. I'm trying to aim my blog posts at a somewhat less advanced audience, seeing as how there's a plethora of good advanced content available already.

Luke said...

I understand. I think the thing I really would have liked to have seen earlier on is the emphasis on vectors being dependent on co-ordinate systems. Whenever one writes down a vector, one immediately has to ask, with respect to what basis? This is what I never seen in introductions which I think should be mentioned. In quantum mechanics it's there but not so much talked about explicitly.

Luke said...

Glad to see it was emphasised. It certainly never was in my education. My GR prof told us to unlearn everything we've ever learnt about vectors and relearn it in the more advanced way. It was only then did I appreciate vectors.

I look forward to more posts. Have you ever read the book by Lecture in Quantum Mechanics - Basic Matters by Englert? It's very much focused on finite dimensional quantum mechanics and introduces all the bra-ket stuff from page 1. It might be an interesting read if you haven't ever seen it.

Kbt Tate said...

vectors are not basis dependent, neither are tensors. Talking about a vector v or a Linear transformation T makes perfect sense without mentioning a basis...in fact any interesting result will not be dependent on choosing a basis.

In DG, elements of the tangent space or cotangent space can be projected onto the co-ordinate basis (\del x_i or dx^i) of a certain co-ordinate chart you are using....but when you use a different coordinate chart only the projection of a given tangent vector onto that basis changes, not the vector itself...this is key to GR.

Luke said...

I apologise for not being clear. I should have said the components of the tensor depend on the co-ordinate system used depend on the basis.