MAT2611 Assignment 5 Solutions 2026

MAT2611 Assignment 5 Solutions 2026
UNISA
Due date: Friday, 28 August 2026

, Question 1

Given:

𝑣1 = (−, 12), 𝑣2 = (2, −3)
𝑇(𝑣1 ) = (1, −, 21), 𝑇(𝑣2 ) = (0, −, 13)

Find:

1. A formula for 𝑇(𝑥 , 𝑦).

2. Use the formula to find 𝑇(2, −1).

Express (𝒙, 𝒚)as a linear combination of 𝒗𝟏 and 𝒗𝟐

Let

(𝑥 , 𝑦) = 𝑎𝑣1 + 𝑏𝑣2 .


Then

(𝑥, 𝑦) = 𝑎(−, 12) + 𝑏(2, −3).


This gives the system

−𝑎 + 2𝑏 = 𝑥
2𝑎 − 3𝑏 = 𝑦.

From the first equation,

𝑎 = 2𝑏 − 𝑥.

Substitute into the second equation:

2(2𝑏 − 𝑥) − 3𝑏 = 𝑦
4𝑏 − 2𝑥 − 3𝑏 = 𝑦
𝑏 = 𝑦 + 2𝑥.

Then

𝑎 = 2(𝑦 + 2𝑥) − 𝑥 = 2𝑦 + 4𝑥 − 𝑥 = 3𝑥 + 2𝑦.

Hence

(𝑥 , 𝑦) = (3𝑥 + 2𝑦)𝑣1 + (2𝑥 + 𝑦)𝑣2 .
Use the linearity of 𝑻

Since 𝑇is linear,

𝑇(𝑥 , 𝑦) = (3𝑥 + 2𝑦)𝑇(𝑣1 ) + (2𝑥 + 𝑦)𝑇(𝑣2 ).