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While going through the comments on an interesting topic on MathOverflow, I came a cross a quote:

Take three distinct lines in R^2 as U, V, W. All intersections have 0 dimensions.

I have only recently been introduced to the notion of vector spaces, and the examples of vector spaces that I know of include those such as $n$-dimensional Euclidean spaces.

I understand that the dimension of a vector space refers to the cardinality of the basis - easy enough to understand when working with spaces such as $\mathbb{R}^n$.

However, if an arbitrary line in $\mathbb{R}^2$ is indeed a vector space, what would its dimension be, and why? Similarly, why would the dimension of a coordinate point be zero?

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For a line in $\mathbb R^2$ to be a vector space, it must pass through the origin. If so, it is a vector space with dimension 1. With such a line $V$, we only need to pick one nonzero vector $v\in V$ and then for all other $u\in V$, $u=av$ for some scalar $a$. Thus, $\{v\}$ is a basis. If a line does not pass through the origin, we can still define its dimension by translating it, calculating the dimension, and then translating it back. of course, we will always calculate the dimension of such a line to be 1 as well. As for a point, to be a vector space, it must be at the origin to be a vector space (if it is not, we can do the same translating trick). To find the dimension of the vector space consisting of one point, we would need to find a basis on nonzero vectors. Since there are no nonzero vectors in the space, the basis is empty, its cardinality is zero, and the dimension of the space is also zero.

Plutoro
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  • Thank you! I missed the fact that a line would have to contain the origin for it to be a proper vector space. That said, what you say about translating a space to define the dimension is really interesting. Does that allow for a valid definition of dimension for any vector space that does not contain the zero vector? –  Sep 01 '15 at 15:49
  • If a space does not contain an origin, it is not a vector space. In some cases, such a space may not be describable as a subset $\mathbb R^n$, and then this trick wouldn't work. Other spaces may not have other vector space properties when translated. The only spaces for which this will work is affine spaces, which are subsets of $\mathbb R^n$ (or $ \mathbb C^n$) which look like $\mathbb R^m$ (or $\mathbb C^m$) for $m<n$, but which do not neciessrily pass through the origin. A line is such a space. – Plutoro Sep 01 '15 at 16:11
  • @River: See affine space and Definition of an affine subspace and What are differences between affine space and vector space?. – Dave L. Renfro Sep 01 '15 at 16:11
  • @DaveL.Renfro +AlexS Thank you very much! –  Sep 02 '15 at 02:53