Let y1, y2, and y3 be defined as in Exercise 7, and let L be the linear operator on R3 d...

Let y₁, y₂, and y₃ be defined as in Exercise 7, and let L be the linear operator on R³ defined by

L ( c ₁ y ₁ + c₂y₂ + c₃y₃) = (c₁ + c₂ + c₃)y₁ + (2c₁ + c₃)y₂ − (2c₂ + c₃)y₃

(a) Find a matrix representing L with respect to the ordered basis {y₁, y₂, y₃}.

(b) For each of the following, write the vector x as a linear combination of y₁, y₂, and y₃ 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

Reference: Exercise 7:

Let

and let be the identity operator on R³.

(a) Find the coordinates of with respect to {y₁, y₂, y₃}.

(b) Find a matrix A such that Ax is the coordinate vector of x with respect to {y₁, y₂, y₃}.