MAT2611 ASSIGNMENT 9 2025
Problem 1
𝑣1 = ( −2, 3), 𝑣2 = (−1, 0)
𝑇: ℝ2 → ℝ3
The set 𝑆 is linearly independent and may be used as a basis for ℝ2
Any vector (𝑥, 𝑦) ∈ ℝ2 may uniquely be written as linear combination of 𝑣1 and 𝑣2
(𝑥, 𝑦) = 𝑘1 (−2, 3) + 𝑘2 (−1, 0)
where 𝑘𝑖 ∈ ℝ
(𝑥, 𝑦) = (−2𝑘1 , 3𝑘1 ) + (−𝑘2 , 0)
(𝑥, 𝑦) = (−2𝑘1 − 𝑘2 , 3𝑘1 )
−2𝑘1 − 𝑘2 = 𝑥
3𝑘1 = 𝑦
1
𝑘1 = 𝑦
3
−2𝑘1 − 𝑘2 = 𝑥
1
−2 ( 𝑦) − 𝑘2 = 𝑥
3
2
−𝑘2 = 𝑥 + 𝑦
3
2
𝑘2 = −𝑥 − 𝑦
3
1 2
(𝑥, 𝑦) = 𝑦 (−2, 3) + (−𝑥 − 𝑦) (−1, 0)
3 3
1 2
𝑇(𝑥, 𝑦) = 𝑇 ( 𝑦( −2, 3) + (−𝑥 − 𝑦) ( −1, 0))
3 3
Problem 1
𝑣1 = ( −2, 3), 𝑣2 = (−1, 0)
𝑇: ℝ2 → ℝ3
The set 𝑆 is linearly independent and may be used as a basis for ℝ2
Any vector (𝑥, 𝑦) ∈ ℝ2 may uniquely be written as linear combination of 𝑣1 and 𝑣2
(𝑥, 𝑦) = 𝑘1 (−2, 3) + 𝑘2 (−1, 0)
where 𝑘𝑖 ∈ ℝ
(𝑥, 𝑦) = (−2𝑘1 , 3𝑘1 ) + (−𝑘2 , 0)
(𝑥, 𝑦) = (−2𝑘1 − 𝑘2 , 3𝑘1 )
−2𝑘1 − 𝑘2 = 𝑥
3𝑘1 = 𝑦
1
𝑘1 = 𝑦
3
−2𝑘1 − 𝑘2 = 𝑥
1
−2 ( 𝑦) − 𝑘2 = 𝑥
3
2
−𝑘2 = 𝑥 + 𝑦
3
2
𝑘2 = −𝑥 − 𝑦
3
1 2
(𝑥, 𝑦) = 𝑦 (−2, 3) + (−𝑥 − 𝑦) (−1, 0)
3 3
1 2
𝑇(𝑥, 𝑦) = 𝑇 ( 𝑦( −2, 3) + (−𝑥 − 𝑦) ( −1, 0))
3 3
