Why is that in n dimensions, we can't represent lower dimensional entities with a single linear equation in cartesian coordinates?

Question

In one dimension, we can represent a point as the solution set of a single algebraic equation.

In two dimensions, we can represent a line through a single equation, and points as the common solution set of certain lines.

In three dimensions, we can represent plane as a single equaiton, lines as common solutions of planes, and points as intersection of three planes.

It seems so in n dimensions, we can not represent an n-2 dimensional object (and even anything lower than that) with a single equation. Why is that?

Mildly related

Before attempting to answer this we'd need to know why you think we can't? — ancient mathematician, Mar 27 '24 at 07:36
You must have some reason for asserting such an unlikely thing. — ancient mathematician, Mar 27 '24 at 07:39
I think you didn't understand ancientmathematician comment. We can represent an $n-1$ dimensional objects with a single equation. — jjagmath, Mar 27 '24 at 07:40
So, could you explain represent how to write a line in three dimensions with only a single equation? — tryst with freedom, Mar 27 '24 at 07:40
For $n = 3$, $n - 1 = 2$. A line is not a 2-dimensional object. — paste bee, Mar 27 '24 at 07:51
It depends on what you mean by an "equation" and "entity." Your tag is "linear algebra", do you mean "linear equation" and "linear subspaces" (for "entities")? — Moishe Kohan, Mar 27 '24 at 08:18
In linear algebra classes one typically learns the Rank-nullity theorem, which answers your quesion... — Moishe Kohan, Mar 27 '24 at 08:28

score 1 · Answer 1 · answered Mar 27 '24 at 07:49

1

A single equation $$|x+3|+|5-y|+|z-1|=0$$ represents a $0$-dimensional point $(-3, 5, 1)$ in $\mathbb R^3.$

answered Mar 27 '24 at 07:49

CiaPan

13,049

$n$ will be at least four. – Bob Dobbs Mar 27 '24 at 07:51
@BobDobbs How does it matter? The solution of my equation is three dimensions 'smaller' than the space in which I define it. What is so special in $n$ being 'at least four' which can't be handled the same way? – CiaPan Mar 27 '24 at 08:38
Oh. Sorry. I misunderstood the question first. Then somehow I thought it is about $n\geq4$. @CiaPan – Bob Dobbs Mar 27 '24 at 08:58

Why is that in n dimensions, we can't represent lower dimensional entities with a single linear equation in cartesian coordinates?

1 Answers1