This was in Armstrong's Basic Topology text, p119,
Let $v_0, \ldots, v_k$ be points of euclidean $n$-space $E^n$. The hyperplane spanned by these points consists of all linear combinations $\lambda_0 v_0+ \lambda_1v_1 + \cdots + \lambda_k v_k$, where each $\lambda_i$ is a real number and the sum of the $\lambda_i$ equals $1$.
How did this definition come up?
Aside: I googled and got quite confused with a number of definitions. Particularly for vector spaces:
(i) Let $f:X \rightarrow \mathbb{R}$ be a linear functional on real vector space $X$, then $M_a = \{x : f(x) = a \}$ are the hyperplanes.
(ii) The hyperplanes of a real vector space are subspaces of codimension $1$.
But doesn't (i) and (ii) contradict? (ii) requires hyperplane to be subspaces(?) Whilst $M_a$ are not necessarily subspace as it doesn't have to contain $0$ ... but I do see it is a shift of $M_0$ which is a subspace.
One thing that happens when you google around is words get mixed and matched in funny ways.
If, in (ii), the word "subspace" is interpreted as "vector subspace" then, just as you say, (ii) is not equivalent to (i). If on the other hand "subspace" in (ii) is interpreted as "affine subspace" then (ii) is indeed equivalent to (i).
If, in the quote from Armstrong, we have $k=n-1$ then that definition is equivalent to (i) (and to (ii) with "subspace" interpreted as "affine subspace). For general values of $k$, what is defined in the quote from Armstrong is an "affine subspace of dimension $k$", which by definition is a subset of the form $V+a$ where $V$ is a vector subspace of dimension $k$ and $a$ is some other vector.