Chapter 2.4, Problem 104E

The color of light can be represented in a vector $\left[\begin{array}{l}R\\ G\\ B\end{array}\right]$ , where $R=\text{amount}\text{of}\text{red},G=\text{amount}\text{of}\text{green}$ , and $B=\text{amount}\text{of}\text{Blue}$ . The human eye and the brain transform the incoming signal into the signal $\left[\begin{array}{l}I\\ L\\ S\end{array}\right]$ , where
intensity $I=\frac{R+G+B}{3}$
long-wave signal $L=R-G$
short-wave signal $S=B=\frac{R+G}{2}$
a. Find the matrix P representing the transformation from $\left[\begin{array}{l}R\\ G\\ B\end{array}\right]$ to $\left[\begin{array}{l}I\\ L\\ S\end{array}\right]$ .
b. Consider a pair of yellow sunglasses for water sports that cuts out all blue light and passes all red and green light. Find the $3\times 3$ matrix A that represents the transformation incoming light undergoes as it passes through the sunglasses. All the entries of your matrix A will be 0’s and 1’s.
c. Find the matrix for the composite transformation that light undergoes as it first passes through the sun glasses and then the eye.
d. As you put on the sunglasses, the signal you receive (intensity, long- and short-wave signals) undergoes a transformation. Find the matrix M of this transformation. Feel free to use technology.