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## Main Question or Discussion Point

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.

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.