The dimension of a point and how to construct it

Question

Wikipedia states:

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

Furthermore about the dimension of a point

The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting of a single point (which must be the zero vector 0), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero: $1 \cdot \mathbf{0}=\mathbf{0}$

How would a mathmatician explain how to construct a point, for example $p_1=(1,2,3)\in \mathbb{R}^3$?

I'm told this point has no dimensions, but then there has to be a space that constructs it, a vector space that is. And this space has to have some linearly independent subset, surely I can't construct it from the empty set. I don't know if I can even talk about a dimension of a point which is not zero, since it is no vector space.

Or are points just "there" devoid of any need to be constructed? Somewhere my thinking must be wrong, so clarification is appreciated.

Answer 1

What do you mean by "a space that constructs it"? That might be the source of your confusion.

$(1,2,3)\in \mathbb{R^3}$. If you want a vector space that has $(1,2,3)$ in it, that's a good answer. If you want to show someone that point, I would draw $\mathbb{R}^3$ and put a dot at the coordinates $(1,2,3)$.

If you want a vector space generated by $(1,2,3)$, you get a copy of $\mathbb{R}$ because you get all of its scalar multiples. As a subspace of $\mathbb{R}^3$ this vector space would be $\{(x,y,z):6x=3y=2z\}$

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Re: the dimension of $(1,2,3)$

There are several ways to measure dimension. $(1,2,3)$ has a geometric dimension of $0$ because it's a point, but in the sense of dimension that you quote it doesn't have dimension because it's not a vector space, and the vector-space dimension (which is what that definition is) is only defined for vector spaces.

The most common notion of geometric dimension (which is what we talk about when we say a triangle is 2D) comes from embedding geometric objects inside of vector spaces, and assigning them the dimension equal to the minimum $k$ such that they embed into $\mathbb{R}^k$. So a single point embed into the $0$-dimensional vector space $\{0\}~\mathbb{R}^0$ and therefore is zero dimensional. A triangle embeds into $\mathbb{R}^2$ but does not embed into $\mathbb{R}^1$ and is therefore one dimensional. Generalizing this definition out of geometry and into topology gives rise to the Lebesgue Dimension, also known as the topological dimension. The Lebesgue dimension agrees with the embedding dimension, but allows you to talk about the dimension of topological objects more generally.