A hyperplane is a subspace whose dimension is one less than that of its ambient space. Why?

Question

If a space is $3$-dimensional then its hyperplanes are the $2$-dimensional planes, while if the space is $2$-dimensional, its hyperplanes are the $1$-dimensional lines. why? furthermore, how does a line exist in $3$d.

Answer 1

The simple answer is that is just the definition. However, I think a little bit can be said about why the name is reasonable.

First note that "hyper" is at sometimes used as a prefix for generalizations to higher dimensions of three dimensional objects. E.g. a hypercube is the generalization of a cube to n dimensions. So "hyperplane" should be taken to mean "the generalization of a plane in n dimensional space".

The question then becomes, "what is the right way to generalize a plane to n dimensional space?" There are two obvious answers. A plane is a subspace of $\mathbb{R}^3$ which has dimension $2$, so its reasonable to say a plane in $\mathbb{R}^n$ is a subspace of dimension $2$. This isn't really a high dimensional generalization though- it comes out of viewing $\mathbb{R}^3$ a subspace of $\mathbb{R}^n$.

Another way to think of a plane is that it is the set of vectors in $\mathbb{R}^3$ which are orthogonal to a given nonzero vector. Generalizing this definition leads to hyperplanes. That is, a hyperplane in $\mathbb{R}^n$ is the set of vectors which is orthogonal to some fixed nonzero vector in $\mathbb{R}^n$.

As for why a hyperplane has dimension $n-1$ instead of $n$, well with the above definition it cannot have dimension $n$ otherwise your fixed vector would be orthogonal to itself. Probably the easiest way to show that the hyperplane in fact has dimension $n-1$ and not smaller is to use the fact that any vector can be extended to an orthogonal basis. The $n-1$ vectors you have added form an orthogonal basis for the hyperplane.

Answer 2

Because that's the definition. Let $V$ be a vector space and let $n$ be the dimension of $V$. Then a subspace $W$ of $V$ is called a hyperplane of $V$ if the dimension of $W$ is $n-1$.

And to answer your second question, just look at your surroundings. You can surely see shapes that resemble a line and in the same way we can have lines in $\mathbb{R}^3$. For example consider the set of all points $(x, y, z)$ in $\mathbb{R}^3$ where $x = y = z$. This is a line through the origin[$(0, 0, 0)$] if you try to graph it. In general all the lines in $\mathbb{R}^3$ are the points that satisfy a certain equation of the form: $$ \frac{x - x_0}{m} = \frac{y - y_0}{n} = \frac{z - z_0}{l},$$ for some $(x_0, y_0, z_0) \in \mathbb{R}^3$ and $(m, n, l) \in \mathbb{R}^3$.

Answer 3

First, there is not a universal agreement that one cannot use the word "hyperplane" for anything of dimension other than $n-1$ in an $n$-dimensional space. For example, the definition of hyperplane in Linear Algebra by Waldron, Cherney, & Denton allows on to construct a $k$-dimensional hyperplane for any $k\leq n$ in an $n$-dimensional space. According to these authors, it is usual (but not mandatory) to assume the dimension is $n-1$ if the dimension is not explicitly specified.

The reason so many other authors use hyperplane exclusively for a subspace of dimension $n-1$ is because they find the subspaces of dimension $n-1$ especially interesting and useful, so they would like to refer to them frequently, so they would like a convenient term by which to refer to such a thing that is less cumbersome than "subspace of dimension $n-1$".

Since we already have the word space for the $n$-dimensional subspace of an $n$-dimensional space, it would be redundant to also define the same object as a hyperplane. So instead we use hyperplane for another interesting kind of subspace, namely a subspace of $n-1$ dimensions.

By intersecting two hyperplanes of this kind (ones with $n-1$ dimensions) you can get a subspace of dimension $n-2$. (This happens if the hyperplanes are not parallel.) By intersecting that subspace with another hyperplane, it is possible to get a subspace of dimension $n-3.$ (The hyperplane must not be parallel to the $(n-2)$-dimensional space.) Ultimately you can produce subspaces of any dimension less than $n$ by this method, including lines.

A line in three dimensions is the intersection of two non-parallel planes. Alternatively, if you're using vectors, the scalar multiples of a single vector determine a line.