There once was a couple named Bright
Who could make love much faster than light
They started to play
In a relative way
And both came the previous night
So…11d M-Theory would use an 11 vector matrix?
See, now you’re making me pull out the hard answer. Before talking about M-theory, let’s just discuss general relativity. Spacetime in GR is a “locally Lorentzian 4-manifold”. Let’s break this down.
An n-manifold is a topological space that can be covered by a number of patches, each one looking like a blob in R[sup]n[/sup] and fitting together “smoothly” on the overlap. There’s a lot of technicality that gets into what’s meant by “smooth”, but you don’t really need to know the details. A good example to get started is a 2-sphere like (roughly) the surface of the Earth. In fact, this shows how something that looks like R[sup]2[/sup] locally can fail to look like it globally.
At any point on an n-manifold we can define (in various ways) the “tangent space” to the manifold at that point. One nice definition says to consider curves passing through that point and consider two curves equal if they have the same velocity as they pass through the point. It’s pretty easy to see that this is an n-dimensional vector space, just like R[sup]n[/sup] is. In a manifold, though, we give up the ability to slide vectors from point to point like we did in R[sup]n[/sup]. Each tangent vector lives at a definite point. Still, here we have recovered a nice copy of R[sup]4[/sup] associated to each point in spacetime. This is why the easy answer to your original question was “yes”.
By taking all of the tangent spaces and sewing them together “smoothly”, we get the “tangent bundle” of M, written T(M). In a small enough neighborhood U of a point in M, this looks like UxR[sup]n[/sup], but in general the topology can get pretty complicated. T(M) comes with a mapping P:T(M)–>M sending each vector to the point in M where it lives. Now we can talk about vector fields as “sections of the tangent bundle”. That is: a vector field v is a map from M to T(M) such that P(v§)=p for each point p in M. In other words, at each point we pick a vector at that point that varies “smoothly” as we move p.
To say that GR is “locally Lorentzian” is to insist that physics look like special relativity on each tangent space. This thread (up until now) has been concerned with sepcial relativity only, which takes place in R[sup]4[/sup]. Now we say that that’s a description of what goes on in the tangent space at each point. To do this, we specify a metric.
A metric defines a quadratic form on each tangent space, which is a way of getting a “squared-length” for each vector in the space. We also insist that at each point we can transform this quadratic form into the Lorentz form, which was the one I used to calculate squared-lengths in an earlier post in this thread. That is, we can pick three vectors having squared-length 1 in the tangent space and one of squared-length -1 that are all “perpendicular” (a metric gives a way of determining this too). We also insist that though the metric varies from point to point, if we plug in a smooth vector field we should get a smooth function describing its squared-length at each point.
Now, GR is basically the study of the geometry of this sort of 4-manifold. In fact, just last night I wrote up a quick history of the Einstein equation in this thread.
M-theory is insanely more complicated than this. It deals with a more general kind of fiber bundle than T(M), which sort of comes with the manifold. Instead of having a copy of R[sup]n[/sup] at each point, the 10-d superstring theories have a copy of a 6-dimensional manifold F at each point. Around any point p in M we can pick a small enough neighborhood U such that the bundle above U looks like UxF, but in general things get a lot more twisted.
As an example, consider a cylinder. This can be built up out of a circle S and an interval I by putting a copy of the interval on each point of the circle. It’s so simple that the whole thing looks like SxI. Now consider a Möbius strip. Again, you can build it up with a copy of the interval at each point on the circle, but now the interval twists around the circle before joining back up with itself. Around any point of the circle it looks like a rectangle, but globally it doesn’t look like a simple cylinder. This is just a 1-manifold (the interval) fibered over another 1-manifold (the circle). Imagine what can happen with a 6-manifold fibered over a 4-manifold!
Basically, my point is that while even through general relativity we’re still mostly concerned with vectors (which happen to have exactly as many components as spacetime has dimensions), more advanced theories start considering much weirder objects which look nothing at all like vectors.
It’s been said that the more you realize how ignorant you are, the closer you are to wisdom. I feel a lot wiser after reading this thread. Boy howdy! I was thinking that “index juggler” might be a cool job description, but maybe not. 
It was Socrates who said, after one of his famous interrogations, that unlike the wise men he questioned, he knew nothing, and that is more than the others knew because he, at least, knew something: that he knew nothing. I read this thread and don’t feel any wiser at all. 
Probably because I didn’t understand a word of it.
Wow, after my explanation (posting #4), I got kind of lost myself. And I wrote that “Relativity Calculator”!!
Sheesh, I feel like a 6th grader in an advanced doctoral seminar.