Chapter 4.2, Problem 8E

Let $y_1,y_2,$ and $y_3$ be defined as in Exercise 7, and let L be the linear operator on ${ℝ}^3$ defined by

$L(c_1{y}_1+c_2{y}_2+c_3{y}_3)=(c_1+c_2+c_3)y_1+(2c_1+c_3)y_2-(2c_2+c_3)y_3$

(a) Find a matrix representing L with respect to the ordered basis $\left\{y_1,y_2,y_3\right\}$ .

(b) For each of the following, write the vector x as a linear combination of $y_1,y_2,$ and $y_3$ and use the matrix from part (a) to determine $L(x)$ :

(i) $x={(7,5,2)}^T$

(ii) $x={(3,2,1)}^T$

(iii) $x={(1,2,3)}^T$