Exercise 9.1.5 In each case, verify Theorem 9.1.3. Use the standard basis in R^n and {1, x, x^2} in P2.
a. R^3 -> T -> R^2 -> S -> R^4; T(a, b, c) = (a + b, b - c), S(a, b) = (a, b - 2a, 3b, a + b)
b. R^3 -> T -> R^4 -> S -> R^2; T(a, b, c) = (a + b, c + b, a + c, b - a), S(a, b, c, d) = (a + b, c - d)
c. P2 -> T -> R^3 -> S -> P2; T(a + bx + cx^2) = (a, b - c, c - a), S(a, b, c) = b + cx + (a - c)x^2
d. R^3 -> T -> P2 -> S -> R^2; T(a, b, c) = (a - b) + (c - a)x + bx^2, S(a + bx + cx^2) = (a - b, c) kindly do a and c.