# 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. • Check out the following link to help format your question. meta.math.stackexchange.com/questions/5020/… – Vincent Aug 29 '14 at 16:59 • what's$A$? Please define all the terms. – Memming Aug 29 '14 at 17:19 • This looked like a linear programming problem to me, rather than linear algebra... – Diya Aug 29 '14 at 17:32 ## 3 Answers 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. 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} $This sort of thing always confused me until I started thinking of matrices not as things, but as functions. An$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? $\mathbf{A}^T$.

$$\begin{bmatrix}1 & 40 & 2\\50 & 1000 & 50\end{bmatrix}\begin{bmatrix}\700\\\3\\\3000\end{bmatrix} = \begin{bmatrix}\700 + \120 + \6000\\\35000 + \30000 + \150000\end{bmatrix} = \begin{bmatrix}\6820\\\188000\end{bmatrix}$$