Linear Algebra problem (related to transpose matrices) Producing $x_1$ trucks and $x_2$ planes requires $x_1+50x_2$ tons of steel, $40x_1+1000x_2$ pounds of rubber, and $2x_1+50x_2$ months of labor. If the unit costs $y_1, y_2, y_3$ are \$700 per ton, \$3 per pound, and \$3000 per month, what are the values of one truck and one plane? Those are the components of $(A^T)y$.
I don't understand the role of the unit costs $y_1, y_2, y_3$.
How do I set up the system for this problem? And why is it $(A^T)y$?
This problem is from the book Linear Algebra and Its Applications 4ed by Prof Gilbert Strang, page 65, chapter 1.6, problem 62.
 A: Let $A=\begin{bmatrix}1&50\\40&1000\\2&50\end{bmatrix}$.  If $x=\begin{bmatrix}x_1\\x_2\end{bmatrix}$, then $Ax$ gives the amount of resources (steel, rubber, and labor) to build $x_1$ trucks and $x_2$ planes; 
and the first column of $A$ gives the resources required for 1 truck, 
while the second column gives the resources required for 1 plane.
If $y=\begin{bmatrix}700\\3\\3000\end{bmatrix}$, then $y$ gives the unit costs for steel, rubber, and labor; so the entries of
$A^{T}y=\begin{bmatrix}1&40&2\\50&1000&50\end{bmatrix}\begin{bmatrix}700\\3\\3000\end{bmatrix}=\begin{bmatrix}6,820\\188,000\end{bmatrix}$ give the cost to make one truck and one plane.
A: To calculate the costs for one truck, you have to set $x_1$ equal to 1. Then you have to multiply the coefficients for $x_1$ with the costs:
Costs for one truck: $1 \cdot 700+ 40 \cdot 3+2\cdot 3,000=6,820$
Costs for one plane: similar calculation.
The term for calculations can be written by matrices $\left( A^T \right) y$:
$
\begin{pmatrix}1 & 40 & 2  \\50 & 1000 & 50  \\\end{pmatrix}\times 
\begin{pmatrix}
700   \\
3 \\
3000   
\end{pmatrix}
$    
A: This sort of thing always confused me until I started thinking of matrices not as things, but as functions.
An $m \times n$ matrix is a function from $R^n$ to $R^m$, from vectors with $n$ elements to vectors with $m$ elements. In this problem, we start out with the $3 \times 2$ matrix
$$\mathbf{A} = \begin{bmatrix}1 & 50\\ 40 & 1000\\ 2 & 50\end{bmatrix}$$
which, when we multiply it by a vector $x = \begin{bmatrix}\text{number of trucks}\\\text{number of planes}\end{bmatrix} = \begin{bmatrix}x_1\\x_2\end{bmatrix}$, yields:
$$\begin{bmatrix}1 & 50\\ 40 & 1000\\ 2 & 50\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}x_1 + 50x_2\\40x_1 + 1000x_2\\2x_1 + 50x_2\end{bmatrix} = \begin{bmatrix}\text{tons of steel}\\\text{pounds of rubber}\\\text{months of labor}\end{bmatrix}$$
So the matrix $\mathbf{A}$ acted like a function: it took in a vector of quantities in trucks-planes space and output a vector of quantities in steel-rubber-labor space.
(This is the reason why an $m \times n$ matrix times a $n \times 1$ matrix will give an $m \times 1$ matrix: we're translating from $R^n$ to $R^m$).
Okay, so as we've got it right now, $\mathbf{A}$ sends trucks-planes to steel-rubber-labor. What matrix will send steel-rubber-labor (now in \$, but that's just a change of units) to trucks-planes? $\mathbf{A}^T$.
$$\begin{bmatrix}1 & 40 & 2\\50 & 1000 & 50\end{bmatrix}\begin{bmatrix}\$700\\\$3\\\$3000\end{bmatrix} = \begin{bmatrix}\$700 + \$120 + \$6000\\\$35000 + \$30000 + \$150000\end{bmatrix} = \begin{bmatrix}\$6820\\\$188000\end{bmatrix}$$
