Equivalence of Definitions of Change of Basis Matrix

Theorem
Let $R$ be a ring with unity.

Let $G$ be a finite-dimensional unitary $R$-module.

Let $A = \left \langle {a_n} \right \rangle$ and $B = \left \langle {b_n} \right \rangle$ be ordered bases of $G$.

Proof
It will be shown that the two matrices defined are equal column-wise.

Let $\displaystyle b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$ for $i$ ranging from $1$ to $n$, where $c_{i j}$'s are scalars.

The uniqueness of the above expression is justified by Expression of Vector as Linear Combination from Basis is Unique.

Then by definition of coordinate vector, the $i$th column of the matrix defined in definition 1 is:
 * $\begin{bmatrix} c_{i 1} & c_{i 2} & \ldots & c_{in} \end{bmatrix}^T$

We also have:
 * $\displaystyle I_G \left({b_i}\right) = b_i = \sum_{j \mathop = 1}^n c_{i j} a_j$

So by definition of relative matrix, the $i$th column of the matrix defined in definition 2 is:
 * $\begin{bmatrix} c_{i 1} & c_{i 2} & \ldots & c_{in} \end{bmatrix}^T$

The two matrices are equal, so the two definitions are equivalent.