MAT3701
ASSIGNMENT 2
Linear Algebra III
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 6
, SOLUTIONS:
Question 1
(1.1.1) To show 𝑇 is linear, let 𝑓, 𝑔 ∈ 𝑃2 (ℝ) and 𝑐 ∈ ℝ.
𝑇(𝑓 + 𝑔)(𝑥) = 𝑥(𝑓 + 𝑔)(𝑥) + (𝑓 + 𝑔)′ (𝑥) = 𝑥(𝑓(𝑥) + 𝑔(𝑥)) + 𝑓 ′ (𝑥) + 𝑔′ (𝑥)
= (𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)) + (𝑥𝑔(𝑥) + 𝑔′ (𝑥)) = 𝑇𝑓(𝑥) + 𝑇𝑔(𝑥).
𝑇(𝑐𝑓)(𝑥) = 𝑥(𝑐𝑓(𝑥)) + (𝑐𝑓)′ (𝑥) = 𝑐𝑥𝑓(𝑥) + 𝑐𝑓 ′ (𝑥) = 𝑐(𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)) = 𝑐𝑇𝑓(𝑥).
Thus 𝑇 is linear.
(1.1.2) Let 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Then 𝑓 ′ (𝑥) = 2𝑎𝑥 + 𝑏, and
𝑇𝑓(𝑥) = 𝑥(𝑎𝑥 2 + 𝑏𝑥 + 𝑐) + (2𝑎𝑥 + 𝑏) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 2𝑎𝑥 + 𝑏.
Page 2 of 6
ASSIGNMENT 2
Linear Algebra III
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 6
, SOLUTIONS:
Question 1
(1.1.1) To show 𝑇 is linear, let 𝑓, 𝑔 ∈ 𝑃2 (ℝ) and 𝑐 ∈ ℝ.
𝑇(𝑓 + 𝑔)(𝑥) = 𝑥(𝑓 + 𝑔)(𝑥) + (𝑓 + 𝑔)′ (𝑥) = 𝑥(𝑓(𝑥) + 𝑔(𝑥)) + 𝑓 ′ (𝑥) + 𝑔′ (𝑥)
= (𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)) + (𝑥𝑔(𝑥) + 𝑔′ (𝑥)) = 𝑇𝑓(𝑥) + 𝑇𝑔(𝑥).
𝑇(𝑐𝑓)(𝑥) = 𝑥(𝑐𝑓(𝑥)) + (𝑐𝑓)′ (𝑥) = 𝑐𝑥𝑓(𝑥) + 𝑐𝑓 ′ (𝑥) = 𝑐(𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)) = 𝑐𝑇𝑓(𝑥).
Thus 𝑇 is linear.
(1.1.2) Let 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Then 𝑓 ′ (𝑥) = 2𝑎𝑥 + 𝑏, and
𝑇𝑓(𝑥) = 𝑥(𝑎𝑥 2 + 𝑏𝑥 + 𝑐) + (2𝑎𝑥 + 𝑏) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 2𝑎𝑥 + 𝑏.
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