Understanding basis and dimension

[dumbQuestion]

I am really confused about something. I know that if I have a vector space, then the dimension of that vector space is the number of elements in a basis for it. But this brings up some confusing issues for me. For example, if we are looking at the null space of a non-singular, square matrix, we know it is just the trivial subspace, a singleton point, the origin. So this trivial subspace is one dimensional and describes exactly one point. However, say we look at the nullspace of a matrix A and find out there is one free variable in A, leading to a basis with just one element: a non-zero point. Then this is a one dimensional subspace and it is just a line in space (all the multiples of that point). So both a line, as well as a singleton point, are one dimensional? Is that correct? That is really confusing to me. I guess when I think of a "line" I just think of 2 dimensions (the xy plane), when I think of a plane, I think of 3 dimensions. When I think of a point, I think of 1 dimension. etc.

 

[lavinia]

The dimension is the number of non zero vectors in a basis. The zero vector space is zero dimensional

 

[chiro]

Hey dumbQuestion.

A good way to think of the dimension is that it's the smallest number of pieces of information you need to represent something and in a vector space, this means the smallest number of "arrows" or vectors needed to represent things in the set.

So if you have only one point in your set you need absolutely no information whatsoever since it is constant which gives a dimension of zero.

If you have a line that can easily be parameterized then it means you have a one-dimensional space since you only need one parameter to accurately define all items in the space.

In a vector space and in linear spaces in general, we have requirements regarding what the classifications can be and we classify things with regard to linear combinations of vectors and this means if we want to describe something in terms of a linear basis, then we need the minimum number of vectors such that a linear combination of these vectors describes everything in our set and all of this kind of thing is at the heart of both theoretical and practical methods in determining what a valid basis and dimension of some vector space is.

 

[dumbQuestion]

Thank you so much for this explanation. To be honest I did not realize that a vector space that's a singleton point was 0-dimensional. But just out of curiosity, don't you at the very least need the singleton point itself to be able to describe it? Or is it because the only singleton point that's a subspace (of R^n) is the origin, and a basis is all the points needed in addition to the "0 element"? (so in that case, no information is needed in addition to the 0 element because the singleton point being described is the 0 element itself)

 

[chiro]

Remember that a dimension captures variation. If there is no variation, there is no need for any set of variables that capture this variation.

The minimum number of variables you need to describe something in some context (in linear algebra we use linear ways, but in non-linear and more generalized contexts we can use quite a number of ways) is what the dimension is specifying.

The goal of doing this is that if you have a whole heap of data, you want to find the best way to describe it and that means finding the minimum number of variables to do so.

Having the simplest form of something is not only good for mathematical reasons where you can do calculations and other analysis: it's good because when its in the simplest form, you can look at this form and try and make sense of what it actually means.

For example if we got a signal and got a harmonic representation, then we could look at this harmonic representation and see what's going on in the frequency domain.

Similarly we know if we can reduce some n-dimensional representation down to 2 minimal independent variables, then we know that this is basically a deformed surface (like a piece of paper) in the high-dimensional space.

If we looked at the n-dimensional embedded surface as n-dimensional, we would mistake this object to have higher complexity and refer to something else when in fact it's just a piece of paper embedded in an n-dimensional space.

Also recall that in general, people are trying to find patterns and finding patterns usually translates into finding the simplest description of an object because if you have the simplest description, you have a description which describes the best pattern you have for that object.

So we need a formal way of doing this in general and the idea of dimension and basis helps us do exactly this.

 

[dumbQuestion]

Thank you very much for this elaboration, it makes a lot more sense now!