Show that this mapping is a linear transformation Show that the map v = [v]E → [v]B for all v ∈ Rn defines a linear transformation TB : Rn → Rn.
B = {b1, b2, b3,...,bn} is a basis of Rn. Any vector v ∈ Rn can be uniquely expressed as a linear combination of B: v = c1b1 + c2b2 + ... + cnbn. The n-tuple (c1 to cn) is called the representation of v in B, denoted by [v]B. v, a vector in Rn can be viewed as v represented in the standard basis E = {e1, e2,...,en}.
 A: Let $f(\mathbf v_E) = \mathbf v_B$.  Can you show that the map $f$ satisfies the definition of a linear map, i.e. that $f(\mathbf u + \mathbf v) = f(\mathbf u) + f(\mathbf v)$ and that $f(\alpha\,\mathbf u) = \alpha\, f(\mathbf u)$?  (Hint: You should be able to do this just by looking at the definition of $\mathbf v_B$.)

OK, let me unpack that a little:
Given a basis $B = \{\mathbf b_1, \mathbf b_2, \dotsc, \mathbf b_n\}$, let $\mathbf u = c_1 \mathbf b_1 + \dotsb + c_n \mathbf b_n$ and $\mathbf v = d_1 \mathbf b_1 + \dotsb + d_n \mathbf b_n$, so that, by definition, $\mathbf u_B = (c_1, \dotsc, c_n)$ and $\mathbf v_B = (d_1, \dotsc, d_n)$.  From this, it directly follows that $\mathbf u + \mathbf v = (c_1 + d_1) \mathbf b_1 + \dotsb + (c_n + d_n) \mathbf b_n$, and thus that $(\mathbf u + \mathbf v)_B = \mathbf u_B + \mathbf v_B$.
By a similar argument, you can show that $(\alpha\, \mathbf u)_B = \alpha\, (\mathbf u_B)$ for $\alpha \in \mathbb R$.  Together, these results prove that the map $\mathbf v \mapsto \mathbf v_B$ is linear, as the question asks.
(In the question you quoted, the vector $\mathbf v$ is identified with its standard representation $\mathbf v_E = (a_1, \dotsc, a_n)$ given by $\mathbf v = a_1 \mathbf e_1 + \dotsb + a_n \mathbf e_n$, where $E = \{\mathbf e_1, \dotsc, \mathbf e_n\}$ is the standard basis of $\mathbb R^n$.  If you want to distinguish between $\mathbf v$ and $\mathbf v_E$, and specifically show that the map $\mathbf v_E \mapsto \mathbf v_B$ is linear, it's enough to show that $\mathbf v_E \mapsto \mathbf v$ is also (trivially) linear, and then apply the lemma that the composition of two linear maps is linear.)
A: Create a matrix $M$ whose columns are the $b_{i}$, then $T(u)=Mu$ performs the transformation. It is now easy to show that $T$ has the properties of a linear transformation.
