Problem II Consider a linear transformation L₁ R4 → R³ given below and a linear operator L₂ R³ R³ given by L₂(x, y, z) = (x + 2y — 3z, 2x + 5y + z, 3x + 8y + 5z). Let L = L₂0 L₁ be their composition, i.e., L(v) = L₂(L₁(v)) for all v € R¹. - (i) Find the matrix of the linear transformation L. (ii) Find a basis for the range of L. (iii) Find a basis for the kernel of L. 21. L₁ (U1, U2, U3, U4) = (v₁ +2V2, V4, Us). 22. L₁(U1, U2, U3, U4) = (U3, 202 - V4, V₁). 23. L₁(U1, U2, U3, U4) = (v2 - 2V3, V4, V₁). 24. L₁ (U1, U2, U3, U4) = (v2V3, V1, 204). 25. L₁ (U1, U2, U3, U4) = (V₁, V₁ + V2, Vs). 26. L₁(U1, U2, U3, U4) = (U₁ +2U3, U4, U2). 27. L₁ (U1, U2, U3, U4) = (303 — 204, U2, U₁). 28. L₁(U1, U2, U3, U4) = (V₁, V₁ + V3, V₂). 29. L₁(U₁, U2, U3, U₁) = (V3 - 2V4, V₁, V₂). 30. L1 (U1, U2, U3, U4) = (V₁ + V₁, V2₂ +V4, V3). 31. L₁(v1, U2, U3, U₁) = (v₁, 202 + V₁, V3). 32. L₁(v1, U2, U3, U4) = (U3, 2V2, V₁ — V₁). 33. L₁(V1, V2, V3, V4) = (V2, V4, V1 - 203). 34. L₁(V1, V2, V3, V4) = (V2, V₁ — U3, 204). 35. L₁(V₁, V2, V3, V4) = (V₁, V2, V₁ + V3). 36. L₁(V₁, V2, U3, U4) (203, U4, U1+U₂). 37. L₁(v1, U2, U3, U4) 38. L₁(v₁, U2, U3, U4) - (303, 02204, V₁). = (V4, V3, V₁ + V₂). 39. L₁(V₁, V2, V3, V₁) = (V3, V₁, V₂ - 20₁). 40. L₁(V1, V2, V3, V₁) = (V₁ — V₁, V2, V3). =
Problem II Consider a linear transformation L₁ R4 → R³ given below and a linear operator L₂ R³ R³ given by L₂(x, y, z) = (x + 2y — 3z, 2x + 5y + z, 3x + 8y + 5z). Let L = L₂0 L₁ be their composition, i.e., L(v) = L₂(L₁(v)) for all v € R¹. - (i) Find the matrix of the linear transformation L. (ii) Find a basis for the range of L. (iii) Find a basis for the kernel of L. 21. L₁ (U1, U2, U3, U4) = (v₁ +2V2, V4, Us). 22. L₁(U1, U2, U3, U4) = (U3, 202 - V4, V₁). 23. L₁(U1, U2, U3, U4) = (v2 - 2V3, V4, V₁). 24. L₁ (U1, U2, U3, U4) = (v2V3, V1, 204). 25. L₁ (U1, U2, U3, U4) = (V₁, V₁ + V2, Vs). 26. L₁(U1, U2, U3, U4) = (U₁ +2U3, U4, U2). 27. L₁ (U1, U2, U3, U4) = (303 — 204, U2, U₁). 28. L₁(U1, U2, U3, U4) = (V₁, V₁ + V3, V₂). 29. L₁(U₁, U2, U3, U₁) = (V3 - 2V4, V₁, V₂). 30. L1 (U1, U2, U3, U4) = (V₁ + V₁, V2₂ +V4, V3). 31. L₁(v1, U2, U3, U₁) = (v₁, 202 + V₁, V3). 32. L₁(v1, U2, U3, U4) = (U3, 2V2, V₁ — V₁). 33. L₁(V1, V2, V3, V4) = (V2, V4, V1 - 203). 34. L₁(V1, V2, V3, V4) = (V2, V₁ — U3, 204). 35. L₁(V₁, V2, V3, V4) = (V₁, V2, V₁ + V3). 36. L₁(V₁, V2, U3, U4) (203, U4, U1+U₂). 37. L₁(v1, U2, U3, U4) 38. L₁(v₁, U2, U3, U4) - (303, 02204, V₁). = (V4, V3, V₁ + V₂). 39. L₁(V₁, V2, V3, V₁) = (V3, V₁, V₂ - 20₁). 40. L₁(V1, V2, V3, V₁) = (V₁ — V₁, V2, V3). =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please solve parts 25-28 of this problem... take as much time as you need.
Transcribed Image Text:Problem II Consider a linear transformation L₁ R4 → R³ given below and a linear
operator L₂ R³ R³ given by L₂(x, y, z) = (x + 2y — 3z, 2x + 5y + z, 3x + 8y + 5z). Let
L = L₂0 L₁ be their composition, i.e., L(v) = L₂(L₁(v)) for all v € R¹.
-
(i) Find the matrix of the linear transformation L.
(ii) Find a basis for the range of L.
(iii) Find a basis for the kernel of L.
21. L₁ (U1, U2, U3, U4) = (v₁ +2V2, V4, Us).
22. L₁(U1, U2, U3, U4) = (U3, 202 - V4, V₁).
23. L₁(U1, U2, U3, U4) = (v2 - 2V3, V4, V₁).
24. L₁ (U1, U2, U3, U4) = (v2V3, V1, 204).
25. L₁ (U1, U2, U3, U4) = (V₁, V₁ + V2, Vs).
26. L₁(U1, U2, U3, U4) = (U₁ +2U3, U4, U2).
27. L₁ (U1, U2, U3, U4) = (303 — 204, U2, U₁).
28. L₁(U1, U2, U3, U4) = (V₁, V₁ + V3, V₂).
29. L₁(U₁, U2, U3, U₁) = (V3 - 2V4, V₁, V₂).
30. L1 (U1, U2, U3, U4) = (V₁ + V₁, V2₂ +V4, V3).
31. L₁(v1, U2, U3, U₁) = (v₁, 202 + V₁, V3).
32. L₁(v1, U2, U3, U4) = (U3, 2V2, V₁ — V₁).
33. L₁(V1, V2, V3, V4) = (V2, V4, V1 - 203).
34. L₁(V1, V2, V3, V4) = (V2, V₁ — U3, 204).
35. L₁(V₁, V2, V3, V4) = (V₁, V2, V₁ + V3).
36. L₁(V₁, V2, U3, U4) (203, U4, U1+U₂).
37. L₁(v1, U2, U3, U4)
38. L₁(v₁, U2, U3, U4)
-
(303, 02204, V₁).
= (V4, V3, V₁ + V₂).
39. L₁(V₁, V2, V3, V₁) = (V3, V₁, V₂ - 20₁).
40. L₁(V1, V2, V3, V₁) = (V₁ — V₁, V2, V3).
=
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