My textbook says the following:

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> Consider a set of basis vector $\mathbf{e}\_i$, $i = 1, \dots, 3$ for a 2-dimensional projective space $\mathbb{P}^2$. For reasons to become clear, we will write the indices as subscripts. With respect to this basis, a point in $\mathbb{P}^2$ is represented by a set of coordinates $x^i$, which represents the point $\sum\_{i = 1}^3 \mathbf{e}\_i$. We write the coordinates with an upper index, as shown. Let $\mathrm{\mathbf{x}}$ represent the triple of coordinates, $\mathrm{\mathbf{x}} = (x^1, x^2, x^3)^T$.
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My understanding from linear algebra is that a basis is a linearly independent spanning set. We then define the dimension of a vector space to be equal to the number of vectors in a basis for said vector space. Therefore, a vector space can have multiple different bases, but each basis must contain the same number of vectors, since this defines the dimension of the vector space. 

So how is it that in the above excerpt the author says that the 2-dimensional projective space $\mathbb{P}^2$ has the basis with *three* vectors $\mathbf{e}\_i$, $i = 1, \dots, 3$? Shouldn't the basis consist of only 2 vectors, since this is a 2-dimensional vector space?

I would greatly appreciate it if people could please take the time to clarify this.