Different use of a matrix and its transpose

We put the main result first, with commentary below.

\[ \target { \left( \leftadj { \left( M^{\source v}_{\target w} (F) = A = \begin{pmatrix} {} a_{\green1\blue1} & a_{\green1\blue2} & \dots & a_{\green1\blue n} \\ && \vdots \\ a_{\green m\blue1} & a_{\green m\blue2} & \dots & a_{\green m\blue n} \\ \end{pmatrix} \right) } \source { \begin{pmatrix} {} x_1 \\ \vdots \\ x_n \\ \end{pmatrix} } \right) X_w\inv = \begin{Bmatrix} {}  ( w_1 \leftadj a_{1,1} + \dots + w_m \leftadj a_{m,1} & \xlongequal[\text{basis}] w & \source{v_1} \leftadj F ) \source{x_1}  \\ & \vdots \\ ( w_1 \leftadj a_{1,n} + \dots + w_m \leftadj a_{m,n} & \xlongequal[\text{basis}] w & {\source{v_n}} \leftadj F ) \source{x_n} \\ \end{Bmatrix}  } \leftadj { \xlongequal[\text {linear}]F } \source { \left( \begin{Bmatrix} {} v_1 x_1 \\ \vdots \\ v_n x_n \\ \end{Bmatrix}  \xlongequal[\text{basis}]v v \right) } \leftadj F = \source v \leftadj F \]

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

\[ \source{ (v_1x_1 + \dots + v_nx_n)\leftadj F \mathrel{\target =} v\leftadj F } \; .  \]