Problem II Consider a linear transformation L₁ : Rª → 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(V1, U2, U3, U4) = (v₁ +2V2, U4, U3). 22. L₁(V1, U2, U3, U4) = (U3, 2U2 - V4, U₁). 23. L₁(V1, U2, U3, U4) = (2-2V3, U4, U1). 24. L₁(V1, U2, U3, U4) = (v2V3, V₁, 204). 25. L₁(V1, U2, U3, U₁) = (V₁, V₁ + V2, 23). 26. L₁(v1, U2, U3, U₁) = (V₁ + 2V3, V4, U2). 27. L₁(U1, U2, U3, U4) = (303-2V4, U2, U₁). 28. L₁ (U1, U2, U3, U4) = (V4, V1 + V3, 12). 29. L₁ (₁, U₂, U3, U₁) = (V3 - 2V₁, V₁, V₂). 30. L₁ (V₁, U2, U3, U4) = (V₁ + V₁, V₂ +04, 03). 31. L₁(V₁, V2, V3, V₁) = (v₁, 2v2 + V₁, V3). 32. L₁(v₁, U2, U3, U4) = (U3, 2V2, V1 — V4). 33. L₁(v1, 12, 13, V4) = (V2, V4, V1 - 203). 34. L₁(V₁, V2, V3, V4) = (V2, V1 — V3, 2v4). 35. L₁(V₁, V2, V3, V4) = (V₁, V2, V₁ + V3). 36. L₁(V1, V2, V3, V4) = (2V3, V4, V₁ + 1₂). 37. L₁(V1, V2, V3, V4) = (3V3, V2 — 204, v₁). 38. L₁(V1, V2, V3, V4) = (V₁, V3, V₁ + 2₂). 39. L₁(V₁, V₂, V3, V₁) = (V3, V₁, V₂ - 20₁). 40. L₁(V₁, V2, V3, V4) = (V1 — V₁, V2, 13).
Problem II Consider a linear transformation L₁ : Rª → 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(V1, U2, U3, U4) = (v₁ +2V2, U4, U3). 22. L₁(V1, U2, U3, U4) = (U3, 2U2 - V4, U₁). 23. L₁(V1, U2, U3, U4) = (2-2V3, U4, U1). 24. L₁(V1, U2, U3, U4) = (v2V3, V₁, 204). 25. L₁(V1, U2, U3, U₁) = (V₁, V₁ + V2, 23). 26. L₁(v1, U2, U3, U₁) = (V₁ + 2V3, V4, U2). 27. L₁(U1, U2, U3, U4) = (303-2V4, U2, U₁). 28. L₁ (U1, U2, U3, U4) = (V4, V1 + V3, 12). 29. L₁ (₁, U₂, U3, U₁) = (V3 - 2V₁, V₁, V₂). 30. L₁ (V₁, U2, U3, U4) = (V₁ + V₁, V₂ +04, 03). 31. L₁(V₁, V2, V3, V₁) = (v₁, 2v2 + V₁, V3). 32. L₁(v₁, U2, U3, U4) = (U3, 2V2, V1 — V4). 33. L₁(v1, 12, 13, V4) = (V2, V4, V1 - 203). 34. L₁(V₁, V2, V3, V4) = (V2, V1 — V3, 2v4). 35. L₁(V₁, V2, V3, V4) = (V₁, V2, V₁ + V3). 36. L₁(V1, V2, V3, V4) = (2V3, V4, V₁ + 1₂). 37. L₁(V1, V2, V3, V4) = (3V3, V2 — 204, v₁). 38. L₁(V1, V2, V3, V4) = (V₁, V3, V₁ + 2₂). 39. L₁(V₁, V₂, V3, V₁) = (V3, V₁, V₂ - 20₁). 40. L₁(V₁, V2, V3, V4) = (V1 — V₁, V2, 13).
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 do parts 20-24... take as much time as needed
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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