MAT2611 Assignment NINE 2023

MAT2611 ASSIGNMENT 9 2023

Problem 31

𝑆 = {𝑣1 , 𝑣2 } is a basis for ℝ2

Where 𝑣1 = (1, −1) and 𝑣2 = (2, 0)

𝑇: ℝ2 → ℝ2 is a linear operator such that:

𝑇(𝑣1 ) = (3,2) and 𝑇(𝑣2 ) = (−1, 0)

To write the vector (𝑥, 𝑦) ∈ ℝ2 in terms of the basis 𝑆

(𝑥, 𝑦) = 𝑘1 (1, −1) + 𝑘2 (2, 0)

(𝑥, 𝑦) = (𝑘1 , −𝑘1 ) + (2𝑘2 , 0)

(𝑥, 𝑦) = (𝑘1 + 2𝑘2 , −𝑘1 )

𝑥 = 𝑘1 + 2𝑘2

𝑦 = −𝑘1

Therefore,

𝑘1 = −𝑦

𝑥 = 𝑘1 + 2𝑘2

𝑥 = −𝑦 + 2𝑘2

1
𝑘2 = (𝑥 + 𝑦)
2

(𝑥, 𝑦) = 𝑘1 (1, −1) + 𝑘2 (2, 0)

1
(𝑥, 𝑦) = −𝑦(1, −1) + (𝑥 + 𝑦)(2, 0)
2

Note that since 𝑇 is a linear operator on ℝ2 , it satisfies the following properties:

For 𝑘 ∈ ℝ and 𝑢, 𝑣 ∈ ℝ2

𝑇(𝑢 + 𝑣) = 𝑇(𝑢) + 𝑇(𝑣) (1)

𝑇(𝑘𝑢) = 𝑘𝑇(𝑢) (2)
1
(𝑥, 𝑦) = −𝑦(1, −1) + (𝑥 + 𝑦)(2, 0)
2

1
𝑇(𝑥, 𝑦) = 𝑇 (−𝑦(1, −1) + (𝑥 + 𝑦)(2, 0))
2