We want to establish a string of equalities.
That string will use several substitutions which involve basis expansions,
so it is helpful to state those basis expansions first.
Since $\source{(v_j)_{1\le j\le n}}$ is a basis for $\source V$, there exists $\source{ \boxed{ (x_j)_{1\le j\le n} } }$ such that
\[ \source{ v \xlongequal[ \text{in terms of the $(v_j)_{1\le j\le n}$ basis of $V$} ] { \text{expanding $v\in V$} } v_1x_1 + \dots + v_nx_n } \; . \]
Since $\target{(w_i)_{1\le i\le m}}$ is a basis for $\target W$, there exists $\leftadj{ \boxed{ (a_{\target i\source j})_{\target{1\le i\le m},\source{1\le j\le n}} } }$ such that
\[ \target{ \source{v_1}\leftadj L \xlongequal[ \text{in terms of the $(w_i)_{1\le i\le m}$ basis of $W$} ] { \text{expanding $\source{v_1}\leftadj L \in W$} } \begin{Bmatrix} {} w_1 \leftadj a_{\target1\source1} \\ \vdots \\ w_m \leftadj a_{\target m\source1} \\ \end{Bmatrix} , \kern2em \ldots \kern2em , \source{v_n}\leftadj L \xlongequal[ \text{in terms of the $(w_i)_{1\le i\le m}$ basis of $W$} ] { \text{expanding $\source{v_n}\leftadj L \in W$ } } \begin{Bmatrix} {} w_1 \leftadj a_{\target1\source n} \\ \vdots \\ w_m \leftadj a_{\target m\source n} \\ \end{Bmatrix} } \]
(Note:
The column vectors of $(\leftadj a_{\target i\source j})_{\target{1\le i\le m},\source{1\le j\le n}}$, written horizontally,
combine to form the transpose of $\leftadj{ M^{\source v}_{\target w} (L) }$.)
So,
\[ \begin{array} {} \target{ \begin{Bmatrix} {} w_1 (\source v \leftadj L)_1 \\ \vdots \\ w_m (\source v \leftadj L)_m \\ \end{Bmatrix} \xlongequal [\text{in terms of the $\target{(w_i)_{1\le i\le m}}$ basis of $W$}] {\text{expanding $\source v\leftadj L \in W$}} \source v\leftadj L \mathrel{\source{ \xlongequal[ \text{in terms of the $\source{(v_j)_{1\le j\le n}}$ basis of $V$} ] { \text{expanding $v\in V$} } }} \source{ ( v_1x_1 + \dots + v_nx_n ) } \leftadj L \mathrel{\leftadj{ \xlongequal[\text{linear}]L} } \source{ v_1} \leftadj L \source{x_1} + \dots + \source{v_n} \leftadj L \source{x_n} = \\ \xlongequal[ \text{in terms of the $\target{(w_i)_{1\le i\le m}}$ basis of $W$} ] { \text{expanding each $\source{v_j}\leftadj L \in W$} } \left( \source{v_1}\leftadj L = \begin{Bmatrix} {} \target{w_1} \leftadj a_{\target1\source1} \\ \target\vdots \\ \target{w_m} \leftadj a_{\target m\source1} \\ \end{Bmatrix} \right) \source{x_1} + \dots + \left( \source{v_n}\leftadj L = \begin{Bmatrix} {} \target{w_1} \leftadj a_{\target1\source n} \\ \target\vdots \\ \target{w_m} \leftadj a_{\target m\source n} \\ \end{Bmatrix} \right) \source{x_n} \\ \xlongequal [\text{dist.}] { \text{dist., Fubini,} }\begin{Bmatrix} {} w_1 \source{(\leftadj A_{\target1} \cdot X)} \\ \vdots \\ w_m \source{(\leftadj A_{\target m} \cdot X)} \\ \end{Bmatrix} } \end{array} \]
\[ \boxed{ \begin{array} {c|ccc} \target{w_1} \source{(\leftadj A_{\target1} \cdot X)} = \target{w_1} \source{ ( \sum_{j=1}^n \leftadj a_{\target1 j} x_j ) } & \target{w_1} \leftadj a_{\target1 \source1} \source{x_1} & \source\dots & \target{w_1} \leftadj a_{\target1\source n} \source{x_n} \\ \target\vdots & \target\vdots & \target\vdots & \target\vdots \\ \target{w_m} \source{(\leftadj A_{\target m} \cdot X)} = \target {w_m} \source{ ( \sum_{j=1}^n \leftadj a_{\target m j} x_j ) } & \target{w_m} \leftadj a_{\target m \source 1} \source{x_1} & \source\dots & \target{w_m} \leftadj a_{\target m\source n } \source{x_n} \\ \hline \\ \begin{array} {} { \displaystyle \mathop{ \target{\sum_{i=1}^m} } \mathop{ \source{\sum_{j=1}^n} } \target{w_i} \leftadj a_{\target i\target j} \source{x_j} } \\ \Vert\rlap{\text{Fubini}} \\ { \displaystyle \mathop{ \source{\sum_{j=1}^n} } \mathop{ \target{\sum_{i=1}^m} } \target{w_i} \leftadj a_{\target i\target j} \source{x_j} } \end{array} & \target{ \begin{Bmatrix} {} w_1 \leftadj a_{1\source1} \\ \target\vdots \\ w_m \leftadj a_{m\source1} \\ \end{Bmatrix} } \source{x_1} & \source\dots & \target{ \begin{Bmatrix} {} w_1 \leftadj a_{1\source n} \\ \target\vdots \\ w_m \leftadj a_{m\source n} \\ \end{Bmatrix} } \source{x_n} \\ & \target\Vert && \target\Vert \\ & \target{ (w \cdot \leftadj A^{\source1}) } \source{x_1} & \source\dots & \target{ (w \cdot \leftadj A^{\source n}) } \source{x_n} \\ \end{array} } \]
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To some extent this follows the notation and setup of Chapter IV, Section 3 of Serge Lang's Linear Algebra, 3e.
The notation deviates from that in these ways:
1. If $f$ is a function and $v$ is in the domain of $f$, we write the value of $f$ at $v$ any of three ways: $f(v)$ or $fv$ or $vf$.
As long as it is known which is a function and which can be an argument to that function, that should not be a problem.
2. Sometimes we have a column of values which need to be added up.
We introduce a notation which gives a vertical version of the standard symbol $\Sigma$ for summation.
For example,
\[ \begin{Bmatrix} {} v_1 \\ \vdots \\ v_n \end{Bmatrix} = \sum_{i=1}^n v_i \; \text{whereas} \begin{pmatrix} {} v_1 \\ \vdots \\ v_n \end{pmatrix} \text{is just a vertically oriented $n$-tuple.} \]
3. Consider the situation:
$f:X\to Y$ is a function and $e$ and $e'$ two expressions denoting the same element of $X$.
We introduce the expression
$(e=e')f$ to
1. Indicate that $e$ and $e'$ denote the same element in $X$, and 2. return the element of $Y$ which is the value of $f$ under $f$.
For a simple example,
\[ 2+3=5 \Rightarrow (2+3)^2=5^2 =25 = (2+3=5)^2 \, . \]
4. For an example, which appeared above, combining these notational conversations, consider
\[ \source { \left( \begin{Bmatrix} {} v_1 x_1 \\ \vdots \\ v_n x_n \\ \end{Bmatrix} \xlongequal[\text{basis}]v v \right) } \leftadj F \; , \]
which follows from
\[ \source{ v_1x_1 + \dots + v_nx_n \xlongequal[\text{basis}]v v } \]
and returns the common value
\[ \source{ (v_1x_1 + \dots + v_nx_n)\leftadj F \mathrel{\target =} v\leftadj F } \; . \]
----------
Here is another approach to the same subject:
\[ \target { \left( \leftadj { \left( M^{\source v}_{\target w} (F) = A =con \begin{pmatrix} {} a_{\green1\blue1} & a_{\green1\blue2} & \dots & a_{\green1\blue n} \\ && \vdots \\ a_{\green m\blue1} & a_{\green m\blue2} & \dots & a_{\green m\blue n} \\ \end{pmatrix} \right) } \source { \begin{pmatrix} {} x_1 \\ \vdots \\ x_n \\ \end{pmatrix} } \right) X_w\inv = \begin{Bmatrix} {} ( w_1 \leftadj a_{1,1} + \dots + w_m \leftadj a_{m,1} & \xlongequal[\text{basis}] w & \source{v_1} \leftadj F ) \source{x_1} \\ & \vdots \\ ( w_1 \leftadj a_{1,n} + \dots + w_m \leftadj a_{m,n} & \xlongequal[\text{basis}] w & {\source{v_n}} \leftadj F ) \source{x_n} \\ \end{Bmatrix} } \leftadj { \xlongequal[\text {linear}]F } \source { \left( \begin{Bmatrix} {} v_1 x_1 \\ \vdots \\ v_n x_n \\ \end{Bmatrix} \xlongequal[\text{basis}]v v \right) } \leftadj F = \source v \leftadj F \]
