: 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+2y3z, 2x + 5y +z, 3x + 8y +52). 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)= (₁ +212, 4, Us). 31. L₁(₁, 2, 3, 4)= (₁, 202 + V₁, Vs).
: 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+2y3z, 2x + 5y +z, 3x + 8y +52). 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)= (₁ +212, 4, Us). 31. L₁(₁, 2, 3, 4)= (₁, 202 + V₁, Vs).
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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31 please with detailed explanation
Transcribed Image Text:Problem II Consider a linear transformation L₁ R¹ R³ given below and a linear
operator L₂ R³ R³ given by L₂(x, y, z) = (x + 2y3z, 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 +202, V4, Us).
22. L₁(U1, U2, U3, U4) = (U3, 202 - V4, U₁).
23. L₁(U1, U2, U3, U4) = (v2 - 2V3, U4, V₁).
24. L1 (U1, U2, U3, U4)= (v2V3, V1, 204).
25. L1 (U1, U2, U3, U4) = (V4, U₁+U2, U3).
26. L1(U1, U2, U3, U4) = (v₁ +2U3, U4, U2).
27. L1 (U1, U2, U3, U4) = (303-204, U2, U₁).
28. L₁(U1, U2, U3, U4) = (V4, V₁ + V3, U₂).
29. L₁(U₁, U2, U3, U4)= (V3-2V4, V₁, V₂).
30. L1 (v1, U2, U3, U4) = (v₁ + V₁, V₂ + V₁, Vs).
31. L₁(v1, U2, U3, U4) = (v₁, 202 + V₁, V3).
32. L₁(v1, U2, U3, U4) = (U3, 202, V1 - V₁).
33. L₁(v1, U2, U3, U4)= (V2, V4, V1 - 203).
34. L₁(v1, U2, U3, U4) (V2, V1-V3, 204).
35. L₁(v1, U2, U3, U4) = (V₁, V2, V₁ + V3).
36. L₁(v₁, U2, U3, U4) (203, U4, U1+U₂).
37. L₁(v1, U2, U3, U4) (303, 02204, ₁).
38. L₁(v1, U2, U3, U4) = (V4, V3, V₁ + V₂).
39. L₁(V₁, V2, V3, V₁) (U3, V₁, V₂ - 20₁).
-
40. L₁(v1, 02, 03, V4) = (V₁ - V₁, V2, V3).
=
=
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