Background question: Suppose $\mathcal{B} = \{v_1, v_2, \dots, v_n\}$ is a basis for an inner product space $V$, and $T: V \to V$ is a linear operator. The goal is to find a matrix representation for $T$ in the basis $\mathcal{B}$, expressed in terms of inner products between the basis vectors $v_1, \dots, v_n$ and also their images $Tv_1, \dots, Tv_n$ under the operator.
Answer to background question: If we let $$ M = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix} $$ be the desired matrix, then for each $i = 1, \dots, n$ it follows that $$ Tv_i = a_{1i} v_1 + a_{2i} v_2 + \dots + a_{ni} v_n $$ and taking an inner product on both sides with $v_j$ yields for each $i,j$ the equation $$ \langle T v_i, v_j \rangle = a_{1i} \langle v_1, v_j \rangle + \dots + a_{ni} \langle v_n, v_j \rangle. $$ Writing these $n^2$ equations in matrix form,
$$ \begin{bmatrix} \langle T v_1, v_1 \rangle & \langle T v_2, v_1 \rangle & \dots & \langle T v_n, v_1 \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle T v_1, v_n \rangle & \dots & \dots & \langle T v_n, v_n \rangle \end{bmatrix} = \begin{bmatrix} \langle v_1, v_1 \rangle & \langle v_2, v_1 \rangle & \dots & \langle v_n, v_1 \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle v_n, v_1 \rangle & \dots & \dots & \langle v_n, v_n \rangle \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & \dots & \dots & a_{nn} \end{bmatrix} $$ Solving yields $$ M = \begin{bmatrix} \langle v_1, v_1 \rangle & \langle v_2, v_1 \rangle & \dots & \langle v_n, v_1 \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle v_n, v_1 \rangle & \dots & \dots & \langle v_n, v_n \rangle \end{bmatrix}^{-1} \begin{bmatrix} \langle T v_1, v_1 \rangle & \langle T v_2, v_1 \rangle & \dots & \langle T v_n, v_1 \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle T v_1, v_n \rangle & \dots & \dots & \langle T v_n, v_n \rangle \end{bmatrix} $$
My question: What to call this procedure/formula, and what (elementary) reference to point students to if they've had one course in linear algebra but haven't seen this. That matrix getting inverted in the answer is the Gramian, and parts of this calculation look like what they might see in their textbook sections on QR decomposition or SVD. Is there something more specific to call this result?