MAT2611
ASSIGNMENT 10
2025
, PROBLEM 1
T1 (x, y) = (4x, x − y, 2x − y)
T2 (x, y, z) = (−x, 0, x + 2y + z)
T3 (x, y, z) = (2x + y − z, 2y + 3z)
(T2 ○ T1 )(x, y) = T2 (T1 (x, y))
= (−(4x), 0,4x + 2(x − y) + (2x − y))
= (−4x, 0,4x + 2x − 2y + 2x − y)
= (−4x, 0,8x − 3y)
(T3 ○ T2 ○ T1 )(x, y) = T3 [(T2 ○ T1 )(x, y)]
= (2(−4x) + 0 − (8x − 3y), 2(0) + 3(8x − 3y))
= (−8x − 8x + 3y, 0 + 24x − 9y)
= (−16x + 3y, 24x − 9y)
PROBLEM 2
1).
T(p(x)) = (p(−1), p(0))
p(x) = 1 + x
∴ p(−1) = 1 + (−1) = 0
∴ p(0) = 1 + 0 = 1
T(1 + x) = (p(−1), p(0))
ASSIGNMENT 10
2025
, PROBLEM 1
T1 (x, y) = (4x, x − y, 2x − y)
T2 (x, y, z) = (−x, 0, x + 2y + z)
T3 (x, y, z) = (2x + y − z, 2y + 3z)
(T2 ○ T1 )(x, y) = T2 (T1 (x, y))
= (−(4x), 0,4x + 2(x − y) + (2x − y))
= (−4x, 0,4x + 2x − 2y + 2x − y)
= (−4x, 0,8x − 3y)
(T3 ○ T2 ○ T1 )(x, y) = T3 [(T2 ○ T1 )(x, y)]
= (2(−4x) + 0 − (8x − 3y), 2(0) + 3(8x − 3y))
= (−8x − 8x + 3y, 0 + 24x − 9y)
= (−16x + 3y, 24x − 9y)
PROBLEM 2
1).
T(p(x)) = (p(−1), p(0))
p(x) = 1 + x
∴ p(−1) = 1 + (−1) = 0
∴ p(0) = 1 + 0 = 1
T(1 + x) = (p(−1), p(0))
