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: 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.
A: 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.
