Q33 please with explanations (linear algebra question) Problem II Consider a linear transformation L₁ R¹ R³ given below and a linearoperator L₂ R³ R³ given by L₂(x, y, z) = (x + 2y3z, 2x + 5y + z, 3x + 8y + 5z). LetL = 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).==

Given: Lv=L2L1v L1v1, v2, v3, v4=v2, v4, v1-2v3L2x, y, z=x+2y-3z, 2x+5y+z, 3x+8y+5z To find: (i) The…

: 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 + 2, 3x + 8y + 5z). Let LL₂0 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. Li (U1, U2, U3, U4)= (1 +202, U4, Us). 22. Li (U1, U2, U3, U4) (Us, 202 U4, U₁). 23. L₁ (U1, U2, U3, U4)= (2-203, U4, U₁). 31. L₁(1, U2, U3, U4) = (v₁, 202 + V4, Vs). 32. Li (U1, U2, U3, U4) = (U3, 202, U₁-V₁). 33. L1 (1, 2, 3, 4) (U2, V4, U1-203). =

: 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 + 2, 3x + 8y + 5z). Let LL₂0 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. Li (U1, U2, U3, U4)= (1 +202, U4, Us). 22. Li (U1, U2, U3, U4) (Us, 202 U4, U₁). 23. L₁ (U1, U2, U3, U4)= (2-203, U4, U₁). 31. L₁(1, U2, U3, U4) = (v₁, 202 + V4, Vs). 32. Li (U1, U2, U3, U4) = (U3, 202, U₁-V₁). 33. L1 (1, 2, 3, 4) (U2, V4, U1-203). =

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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Question

Q33 please with explanations (linear algebra question)

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).
=
=

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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