MAT2611 Assignment 5 Complete Solutions UNISA 2026 LINEAR ALGEBRA

MAT2611
ASSIGNMENT 5
Linear Algebra

FULL
SOLUTIONS
COMPLETE SOLUTIONS

MEMORANDUM
UNISA 2026
Page 1 of 9

, SOLUTIONS

Problem 1

We have 𝑣1 = (−1,2) and 𝑣2 = (2, −3). First we express an arbitrary vector (𝑥, 𝑦) as a
linear combination of 𝑣1 and 𝑣2 .

Let (𝑥, 𝑦) = 𝑎(−1,2) + 𝑏(2, −3). Then:

𝑥 = −𝑎 + 2𝑏, 𝑦 = 2𝑎 − 3𝑏



Solving these equations gives 𝑎 = 3𝑥 + 2𝑦 and 𝑏 = 2𝑥 + 𝑦. Therefore:

(𝑥, 𝑦) = (3𝑥 + 2𝑦)𝑣1 + (2𝑥 + 𝑦)𝑣2



Applying the linear transformation 𝑇:

𝑇(𝑥, 𝑦) = (3𝑥 + 2𝑦)𝑇(𝑣1 ) + (2𝑥 + 𝑦)𝑇(𝑣2 )
= (3𝑥 + 2𝑦)(1, −2,1) + (2𝑥 + 𝑦)(0, −1,3)
= (3𝑥 + 2𝑦, − 6𝑥 − 4𝑦, 3𝑥 + 2𝑦) + (0, − 2𝑥 − 𝑦, 6𝑥 + 3𝑦)
= (3𝑥 + 2𝑦, − 8𝑥 − 5𝑦, 9𝑥 + 5𝑦)



Thus 𝑇(𝑥, 𝑦) = (3𝑥 + 2𝑦, −8𝑥 − 5𝑦, 9𝑥 + 5𝑦).

For (2, −1):

Page 2 of 9