We begin by establishing an invertible linear transformation between any vector space $V$ of dimension $n$ and $\complex{n}$. This will allow us to “go back and forth” between the two vector spaces, no matter how abstract the definition of $V$ might be.
Suppose that $V$ is a vector space with a basis $B=\set{\vectorlist{v}{n}}$. Define a function $\ltdefn{\vectrepname{B}}{V}{\complex{n}}$ as follows. For $\vect{w}\in V$ define the column vector $\vectrep{B}{\vect{w}}\in\complex{n}$ by \begin{align*} \vect{w} &= \vectorentry{\vectrep{B}{\vect{w}}}{1}\vect{v}_1+ \vectorentry{\vectrep{B}{\vect{w}}}{2}\vect{v}_2+ \vectorentry{\vectrep{B}{\vect{w}}}{3}\vect{v}_3+ \cdots+ \vectorentry{\vectrep{B}{\vect{w}}}{n}\vect{v}_n \end{align*}
This definition looks more complicated that it really is, though the form above will be useful in proofs. Simply stated, given $\vect{w}\in V$, we write $\vect{w}$ as a linear combination of the basis elements of $B$. It is key to realize that Theorem VRRB guarantees that we can do this for every $\vect{w}$, and furthermore this expression as a linear combination is unique. The resulting scalars are just the entries of the vector $\vectrep{B}{\vect{w}}$. This discussion should convince you that $\vectrepname{B}$ is “well-defined” as a function. We can determine a precise output for any input. Now we want to establish that $\vectrepname{B}$ is a function with additional properties — it is a linear transformation.
The function $\vectrepname{B}$ (Definition VR) is a linear transformation.
The proof of Theorem VRLT provides an alternate definition of vector representation relative to a basis $B$ that we could state as a corollary (Proof Technique LC): $\vectrepname{B}$ is the unique linear transformation that takes $B$ to the standard unit basis.
Vector representations are most interesting for vector spaces that are not $\complex{m}$.
The function $\vectrepname{B}$ (Definition VR) is an injective linear transformation.
The function $\vectrepname{B}$ (Definition VR) is a surjective linear transformation.
We will have many occasions later to employ the inverse of vector representation, so we will record the fact that vector representation is an invertible linear transformation.
The function $\vectrepname{B}$ (Definition VR) is an invertible linear transformation.
Informally, we will refer to the application of $\vectrepname{B}$ as coordinatizing a vector, while the application of $\ltinverse{\vectrepname{B}}$ will be referred to as un-coordinatizing a vector.