What is an ordered basis? To my understanding, $$e_1 = \{1,0,\ldots,0\},\quad e_2 = \{0,1,\ldots,0\}, \quad \ldots, \quad e_n = \{0,0,\ldots,1\}$$ is an ordered basis for a vector space of dimension $n$. But the group of basis, which implies no ordering is not ordered.

For example: $$\{1,0,\ldots,0\},\quad \{0,1,\ldots,0\},\quad \ldots, \quad\{0,0,\ldots,1\}$$ is not ordered basis, but a basis, for a vector space of dimension $n$.

Is this correct? Thanks!
The difference, of course, is the ordering. An ordered basis $B$ of a vector space $V$ is a basis of $V$ where some extra information is provided: namely, which element of $B$ comes "first", which comes "second", etc. If $V$ is finite-dimensional, one approach would be to make $B$ an ordered $n$-tuple, or more generally, we could provide a total order on $B$.

The difference can be obscured however by common abuse of notation and abuse of terminology regarding bases vs. ordered bases. For example, it would be quite common to say that
$$\{e_1,\ldots,e_n\}$$
is an ordered basis, even though it is just a set and not an ordered $n$-tuple, because the indexes tell you what the intended ordering is.

Let's consider $\mathbb{R}^3$. An example of a plain-old, vanilla basis would be a set
$$\left\{\begin{array}{ccc} &amp; a &amp; \\  &amp; &amp; \alpha\\ \mathbf{a}&amp; &amp; \end{array}\right\}$$
where $a$, $\alpha$, and $\mathbf{a}$ are a basis of $\mathbb{R}^3$ (which I have intentionally written and labeled in a way such that there would be no implied order). An example of an ordered basis could be
$$(a,b,c)$$
which is an ordered $3$-tuple where $a$ comes first, $b$ comes second, and $c$ comes third (reinforced by the fact that that is in alphabetical order), or equivalently, I could have said
$$\{a,b,c\}$$ is an ordered basis where the ordering is specified by $a&lt;b&lt;c$.
A basis is a set of vectors that spans a vector space (or vector subspace), each vector inside can be written as a linear combination of the basis, the scalars multiplying each vector in the linear combination are known as the coordinates of the written vector; if the order of vectors is changed in the basis, then the coordinates needs to be changed accordingly in the new order.

That is the reason why the authors say: "an ordered basis", when they write about, by example, a basis $\{v_1, v_2, \dots, v_n\}$ then it is implicit that an order is said and it is an ordered basis.