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). =

Linear Algebra: A Modern Introduction

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ISBN:9781285463247

Author:David Poole

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Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

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please solve parts 25-28 of this problem... take as much time as you need. Linear algebra review

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).
=

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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