For example, if you calculate the magnitude of a thousand-dimensional vector, you end up with a single scalar. If you calculate the magnitude of a thousand one-dimensional vectors, you'd end up with a thousand scalars.
Additionally, if given a thousand-dimensional vector, the ordering is important, whereas if given a thousand one-dimensional vectors, ordering isn't neccesarily something we know about.
The difference is subtle and occasionally pedantic, but can be very important depending on what exactly one is doing.
Imagine three, 4-by-1 vectors, each "one-dimensional". Twelve total scalars, each vector with four rows and one column. Arrange these three vectors side by side and merge them into a single 4-by-3 matrix. This matrix is "two-dimensional".
Now, let's imagine five such matrices, each 4-by-3. Stack the five matrices one on top of the other. We currently have a 4x3x5 matrix. This matrix, which contains 60 scalars, is "3-dimensional".
Repeat a similar exercise 997 more times and you have a 1000-dimensional matrix.
Compare that matrix to this: 1000 of our original 4-by-1 vectors arranged side by side, which gives a 4x1000 matrix, which is simply a "two-dimensional" matrix with 4000 elements.
A vector with four rows and one column is a four-dimensional vector. A one-dimensional vector can be described with a single number, a 1x1 matrix if you like.
Oops. You're correct: a 2x1 vector is two-dimensional, a 3x1 vector is three-dimensional, etc. <Trying to remember the terminology from linear algebra 15 years ago.> Each element of the mx1 vector represents a magnitude along an orthogonal dimension ('scalars' for a set of 'basis vectors'). So then a 1000x1 vector would be "thousand-dimensional"; each element represents a magnitude along an axis. But is this strictly equivalent to 1000 single-dimensional vectors? `eli173 suggests not, and I agree.
In constructing my incorrect answer in the grand-parent comment, my though process was being guided by the way Matlab/numpy treats these items (and I think I'm on solid ground that Matlab/numpy treat them differently because mathematicians consider them differently). The built-in functions operate very differently (if they work at all) for
size(A) = (m,1)
and
size(A) = (m,n≠1)
So there may be 1000 numbers floating in the ether, but conceptually they're not the same. Multiplying a 1000x1 vector by a 1xp vector has a completely different result than multiplying one thousand 1x1 vectors by that same 1xp vector.
Although, only many hours later do I realize that the original submission title might've been wordplay on the phrase "a picture is worth a thousand words", so my brain is not reliable today. I shall refrain from spewing more-likely-than-not incorrect statements concerning linear algebra.
