Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro, Parker’s, and Bleyer’s. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors 
 $v = \left[\begin{array}{c} 15\cr 52\cr 107 \end{array}\right]$ $p = \left[\begin{array}{c} 2\cr 7\cr 14 \end{array}\right]$ $b = \left[\begin{array}{c} 4\cr 14\cr 29 \end{array}\right]$ Vigoro Parker’s Bleyer’s

Determine the linear transformation $T: \mathbf{R}^3 \rightarrow \mathbf{R}^3$ that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output.
$T(\textbf{x})$ = $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\textbf {x}$
Then find a formula for $T^{−1}$, and use it to determine the amount of Vigoro, Parker’s, and Bleyer’s required to produce $143$ pounds of nitrogen, $497$ pounds of phosphoric acid and $1022$ pounds of potash.
$T^{-1}(\textbf{y})$ = $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\textbf {y}$
$T^{-1}\left(\left[\begin{array}{c} 143\cr 497\cr 1022 \end{array}\right]\right)$ = $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$

You can earn partial credit on this problem.