Problem 2Letbị =b2b3 :and L a linear transformation from R? to R3 given byL(x) = xibı + x2b2 + (¤1 + x2)b3,1.where x =a) Show that {b1, b2, b3} is a basis for R³.b) Find the matrix of L with respect to the bases {e1, e2} and {b1, b2, b3}.

Answered: Problem 2 Let bị = 1 b2 b3 and L a…

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 10 Find the specified change-of-coordinates matrix. Consider two bases B = {b₁, b₂} and C = {₁, c2} for a vector space V such that b₁ = c₁ - 5c2 and by = 2c₁-4c2. Fi coordinates matrix from B to C. O 0 27 -5-4 [12] 024 Question 11
The standard basis & (e₁,e₂) and two custom bases B (b₁,b₂} and C (c₁, c₂} for R² are shown in the figures below. Standard basis S= (₁, ₂) 4 a. Find the change of basis matrix from custom B-coordinates to standard S-coordinates. [id] = id= -2 b Find the change of basis matrix from custom C-coordinates to standard S-coordinates [id] = -1 -1 -1 c. Find the change of basis matrix from B-coordinates to C-coordinates -3 -1/12 1/4 d. Find the change of basis matrix from C-coordinates to B-coordinates. 3 2 1 -1 -2 -9 b2 -3-2-1 82 el bl 1 2 3 b1 X Custom basis B = {b₁,b₂) |id|s [id Standard basis S= (0₁,0₂} a 3 2 1 -1 -2 -9 c2 -3 -2 -1 G2 62 217 el [id] ↑ cl 12 2 3 61 H x Custom basis C (C₁, C₂)
2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with
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Problem 2 Let bị = 1 b2 b3 and L a linear transformation from R? to R³ given by L(r) = 1ibı + x2b2 + (x1 + x2)b3, where x = I2 a) Show that {b1, b2, b3} is a basis for R³. b) Find the matrix of L with respect to the bases {e1, e2} and {b1, b2, b3}.