MAT3701 Assignment 2 Solutions 2026

MAT3701 Assignment 2 Solutions 2026
All questions are answered clearly and in full.


MAT3701 - Linear Algebra III ASSIGNMENT 02
Opens: 04 MAY 2026 Due: 05 JUNE 2026
Instructions for the Assignment
(1) Carefully explain all your arguments.
(2) Only hand written PDF files will be accepted.
(3) Late submissions will not be marked.
(4) Write your name, surname and student number on the first
page.

,MAT3701 – Assignment 02 Solutions

(Write these neatly by hand)

Question 1

Question 1.1

Given the transformation

𝑇: 𝑃2 (ℝ) → 𝑃3 (ℝ)


defined by

𝑇(𝑓(𝑥)) = 𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)


where 𝑃2 (ℝ)is the vector space of all polynomials of degree at most 2.



Question 1.1.1

Show that 𝑇is linear.

A transformation 𝑇is linear if for all polynomials 𝑓(𝑥), 𝑔(𝑥) ∈ 𝑃2 (ℝ)and all scalars 𝑐 ∈
ℝ,

1. 𝑇(𝑓 + 𝑔) = 𝑇(𝑓) + 𝑇(𝑔)

2. 𝑇(𝑐𝑓) = 𝑐𝑇(𝑓)



Let 𝑓(𝑥), 𝑔(𝑥) ∈ 𝑃2 (ℝ).

Then

𝑇(𝑓(𝑥) + 𝑔(𝑥)) = 𝑥(𝑓(𝑥) + 𝑔(𝑥)) + (𝑓(𝑥) + 𝑔(𝑥))′


Using properties of differentiation,

= 𝑥𝑓(𝑥) + 𝑥𝑔(𝑥) + 𝑓 ′ (𝑥) + 𝑔′ (𝑥)


Grouping terms,

= (𝑥𝑓(𝑥) + 𝑓 ′ (𝑥)) + (𝑥𝑔(𝑥) + 𝑔′ (𝑥))


Therefore,

, 𝑇(𝑓 + 𝑔) = 𝑇(𝑓) + 𝑇(𝑔)




Now let 𝑐 ∈ ℝ.

Then

𝑇(𝑐𝑓(𝑥)) = 𝑥(𝑐𝑓(𝑥)) + (𝑐𝑓(𝑥))′
= 𝑐𝑥𝑓(𝑥) + 𝑐𝑓 ′ (𝑥)
= 𝑐(𝑥𝑓(𝑥) + 𝑓 ′ (𝑥))


Hence,

𝑇(𝑐𝑓) = 𝑐𝑇(𝑓)


Therefore 𝑇satisfies both properties of linearity.

Hence,

𝑇 is linear.




Question 1.1.2

Determine a basis for 𝑁(𝑇).

The null space is

𝑁(𝑇) = {𝑓(𝑥) ∈ 𝑃2 (ℝ): 𝑇(𝑓(𝑥)) = 0}


Let

𝑓(𝑥) = 𝑎 + 𝑏𝑥 + 𝑐𝑥 2


Then

𝑓 ′ (𝑥) = 𝑏 + 2𝑐𝑥


Now compute 𝑇(𝑓(𝑥)):

𝑇(𝑓(𝑥)) = 𝑥(𝑎 + 𝑏𝑥 + 𝑐𝑥 2 ) + (𝑏 + 2𝑐𝑥)
= 𝑎𝑥 + 𝑏𝑥 2 + 𝑐𝑥 3 + 𝑏 + 2𝑐𝑥