MAT2611 ASSIGNMENT 5 2025
Problem 1
𝐵 = {1 + 2𝑥, 3 − 𝑥}
𝐵 ′ = {2 − 2𝑥, 4 + 3𝑥}
(b)
Transition from 𝐵 to 𝐵 ′
Polynomials in 𝐵 = {1 + 2𝑥, 3 − 𝑥} need to be written in terms of the polynomials in
𝐵 ′ = {2 − 2𝑥, 4 + 3𝑥}
1 + 2𝑥 = 𝑘1 (2 − 2𝑥) + 𝑘2 (4 + 3𝑥)
1 + 2𝑥 = 2𝑘1 + 4𝑘2 + (−2𝑘1 + 3𝑘2 )𝑥
2𝑘1 + 4𝑘2 = 1 (1)
−2𝑘1 + 3𝑘2 = 2 (2)
(1) + (2) gives:
7𝑘2 = 3
3
𝑘2 =
7
3
2𝑘1 + 4 ( ) = 1
7
12
2𝑘1 = 1 −
7
5
2𝑘1 = −
7
5
𝑘1 = −
14
Problem 1
𝐵 = {1 + 2𝑥, 3 − 𝑥}
𝐵 ′ = {2 − 2𝑥, 4 + 3𝑥}
(b)
Transition from 𝐵 to 𝐵 ′
Polynomials in 𝐵 = {1 + 2𝑥, 3 − 𝑥} need to be written in terms of the polynomials in
𝐵 ′ = {2 − 2𝑥, 4 + 3𝑥}
1 + 2𝑥 = 𝑘1 (2 − 2𝑥) + 𝑘2 (4 + 3𝑥)
1 + 2𝑥 = 2𝑘1 + 4𝑘2 + (−2𝑘1 + 3𝑘2 )𝑥
2𝑘1 + 4𝑘2 = 1 (1)
−2𝑘1 + 3𝑘2 = 2 (2)
(1) + (2) gives:
7𝑘2 = 3
3
𝑘2 =
7
3
2𝑘1 + 4 ( ) = 1
7
12
2𝑘1 = 1 −
7
5
2𝑘1 = −
7
5
𝑘1 = −
14
