MAT2611 ASSIGNMENT 6 2023
Problem 19
𝑇1 (𝑥1 , 𝑥2 ) = (𝑥1 + 𝑥2 , 𝑥1 − 2𝑥2 )
𝑇2 (𝑥1 , 𝑥2 ) = (3𝑥1 , 𝑥1 + 4𝑥2 )
(a)
Let {(1, 0), (0, 1)} be a basis for ℝ2
𝑇1 (1, 0) = (1, 1) = 1(1, 0) + 1(0, 1)
1
Giving the vector [ ]
1
𝑇1 (0,1) = (1, −2) = 1(1, 0) − 2(0, 1)
1
Giving the vector [ ]
−2
The standard matrix for 𝑇1 is:
1 1
[ ]
1 −2
𝑇2 (1, 0) = (3, 1) = 3(1, 0) + 1(0, 1)
3
Giving the vector [ ]
1
𝑇2 (0, 1) = (0, 4) = 0(1, 0) + 4(0, 1)
0
Giving the vector [ ]
4
The standard matrix for 𝑇2 is:
3 0
[ ]
1 4
(b)
The standard matrix for 𝑇1 ∘ 𝑇2 is:
[𝑇1 ][𝑇2 ]
1 1 3 0
=[ ][ ]
1 −2 1 4
Problem 19
𝑇1 (𝑥1 , 𝑥2 ) = (𝑥1 + 𝑥2 , 𝑥1 − 2𝑥2 )
𝑇2 (𝑥1 , 𝑥2 ) = (3𝑥1 , 𝑥1 + 4𝑥2 )
(a)
Let {(1, 0), (0, 1)} be a basis for ℝ2
𝑇1 (1, 0) = (1, 1) = 1(1, 0) + 1(0, 1)
1
Giving the vector [ ]
1
𝑇1 (0,1) = (1, −2) = 1(1, 0) − 2(0, 1)
1
Giving the vector [ ]
−2
The standard matrix for 𝑇1 is:
1 1
[ ]
1 −2
𝑇2 (1, 0) = (3, 1) = 3(1, 0) + 1(0, 1)
3
Giving the vector [ ]
1
𝑇2 (0, 1) = (0, 4) = 0(1, 0) + 4(0, 1)
0
Giving the vector [ ]
4
The standard matrix for 𝑇2 is:
3 0
[ ]
1 4
(b)
The standard matrix for 𝑇1 ∘ 𝑇2 is:
[𝑇1 ][𝑇2 ]
1 1 3 0
=[ ][ ]
1 −2 1 4
