My textbook reads:
"Let $S$ be a basis for a vector space $V$.
For any $\boldsymbol{v}_1,\boldsymbol{v}_2,...,\boldsymbol{v}_r \in V$ and $c_1, c_2, ..., c_r \in \mathbb{R}$,
$(c_1\boldsymbol{v}_1 + c_2\boldsymbol{v}_2+...+c_r\boldsymbol{v}_r)_S=c_1(\boldsymbol{v}_1)_S+c_2(\boldsymbol{v}_2)_S+...+c_r(\boldsymbol{v}_r)_S$"
where $(\boldsymbol{v})_S$ is the coordinate vector of $\boldsymbol{v}$ relative to $S$.
Is there an obvious reason why this is true? I know it is true by writing out both sides explicitly (which was quite tedious), but it seems there is a simple reason why it is true because the textbook calls it a "Remark", not a "Theorem", and just mentions it without proof.
(Note: the textbook defines a vector space as some $V$ such that "either $V=\mathbb{R}^{n}$ or $V$ is a subspace of $\mathbb{R}^{n}$, for some $n$ ", and defines $V$ is to be a subspace of $\mathbb{R}^{n}$ if $V$ is a subset of $\mathbb{R}^{n}$ and $V=span(S)$, where $S=\{u_{1},u_2,...,u_k\} \in \mathbb{R}^{n}$.)
The obvious reason is that the map from $v$ to $(v)_S$ is a homomorphism. Informally, we say that it respects the vector space operations, meaning that $(c·v)_S = c·v_S$ and $(v+w)_S = v_S+w_S$ for any scalar $c$ and vectors $v,w$. You can easily prove these two facts, and hence by the obvious induction you can obtain your textbook's statement.
More abstractly, a homomorphism from a structure to another (in this case from a vector space to another) commutes with the structure's operations. Informally, scale-then-map is the same as map-then-scale, and add-then-map is the same as map-then-add. This is how one should think of homomorphisms.
By the way, your textbook is wrong in its definition of vector space! For example, functions on the real numbers are a vector space over the reals, and it is in no way evident that this vector space has a basis!