# Where can i learn about hyperplanes?

I am learning some theory on convex sets and i need to know what a k-dimensional hyperplane on $$\mathbb{R}^d$$ is. The book i'm using doesn't give a definition (it's a prerequisite for the course), and, despite looking for 2 days straight, i didn't find anything useful on the internet. Where can i find some basic treatment of this topic? (i know the basics of linear algebra and linear transformations)

• A hyperplane in an $n$ dimensional space is an $n-1$ dimensional subspace. Commented Aug 6, 2021 at 16:02
• but the book talks about k-dimensional hyperplanes, what does that mean? Commented Aug 6, 2021 at 17:18
• It likely means a $k$-dimensional subspace. Commented Aug 6, 2021 at 19:05
• It’s unusual to speak of a $k$-dimensional hyperplane in $\mathbb R^d$, unless $k = d-1$. I’d be curious to see a picture of where your book says this. Commented Aug 8, 2021 at 19:13
• The common term I've seen for a $k$-dimensional affine subspace is a $k$-flat. Commented Aug 8, 2021 at 19:25

If you know basic linear algebra then you know about subspaces, and their dimension.

A $$k$$-plane is a translate of a $$k$$-dimensional subspace $$W$$, so a set of the form $$\{ v + w \ | \ w \in W \}$$ for some $$v$$. It's flat like a subspace but need not pass through the origin. Think " plane in space" or "line in space".

The "hyper" in "hyperplane" usually means that $$W$$ is of dimension $$n-1$$ but your book may be using the term for "planes" of any dimension.

Trough every set of $$n$$ vectors in $$\mathbb{R}^{n}$$ passes a halfplane, this halfplane doesn't need to be unique nor doesn't need to intersect the origin $$O \in \mathbb{R}^{n}$$. Every halfplane $$H \subseteq \mathbb{R}^{n}$$ can be described by a non-zero $$a \in \mathbb{R}^{n}$$ and a $$b \in \mathbb{R}$$:

$$H = \{x : x \in \mathbb{R}^{n}, ⟨a, x⟩ = b\}. \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space (1)$$

For $$b$$ we make a distinction:

• $$b = 0$$. This gives $$⟨a, x⟩ = 0$$ and the halfplane $$H$$ passes through $$O$$ and is therefore a vector space. In words $$H$$ are all vectors orthogonal to $$a$$. This is good for your intuition.
• $$b \ne 0$$. The halfplane $$H$$ doesn't go trough $$O$$ and is paralel to the halfplane described by $$⟨a, x⟩ = 0$$ for given $$a$$. This is easy to show because for $$x \in \mathbb{R}^{n}$$ for which holds $$⟨a, x⟩ = 0$$ can not hold $$⟨a, x⟩ \ne 0$$ so $$x$$ can not be contained in the halfplane which doesn't passes trough $$O$$. This shows every halfplane can be described as in (1). Also $$b^{−1}$$ exists so we can write $$⟨b^{−1}a, x⟩ = 1$$ and therefore every halfplane which doesn't intersect $$O$$ can be written as all $$x \in \mathbb{R}^{n}$$ with $$⟨a, x⟩ = 1$$ for an $$a \in \mathbb{R}^{n}$$.

The parameter $$b$$ partitions for a given vector $$a \ne 0$$ the space $$\mathbb{R}^{n}$$ in infinite many halfplanes.