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My book says the following:

$$A_{SS} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix}$$

We have used the subscript $SS$ to indicate that the standard basis is being used to represent the original vectors and also the rotated vectors.

I understand that the standard basis is $e_1 = (1, 0), e_2 = (0, 1)$. What I don't understand is what it means when it mentions that the standard basis is being used to represent the original vectors and also the rotated vectors; in other words, I do not understand the distinction that is being alluded to.

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

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When you have a linear transformation $T$ from a vector space $V$ to a vector space $W$, you can represent that transformation by a matrix, but first you have to pick a basis for $V$ and a basis for $W$. The same is true if $V$ and $W$ are the same vector space – you can use the same basis for both $V$-as-domain and $V$-as-codomain, but you don't have to.

In the example you give, the author is choosing to use the same basis – the standard basis – for the domain and the codomain.

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    Thanks for the response. Can you please produce and example of this distinction? – 2017-02-27
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    I'm not sure what distinction you are referring to. But consider the transformation $T:{\bf R}^2\to{\bf R}^2$ given by $T(v)=2v$. Take $\{\,(2,3),(4,5)\,\}$ as the basis for the domain, and the standard basis for the codomain, then the matrix representing $T$ is $\pmatrix{4&8\cr6&10\cr}$. – 2017-02-27
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    and how would that change if it were the other way around -- if the bases were reversed? – 2017-02-27
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    You tell me. Do you know how to find the matrix representing a linear transformation to given bases? If not, you should post that as a new question (or look around the site to see whether it has been asked and answered previously – I'll bet it has – or sit down with a Linear Algebra textbook, and see how they do it). Comments on an answer, that's not the place to raise new questions. – 2017-02-27