Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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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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