In the vector space R2 consider the two bases given by [u1, u2] with u1 = (1, 2)T, u2 = (2, 5)T and [v1, v2] with v1 = (3, 2)T, v2 = (4, 3)T (both with respect to the standard basis).Follow these steps to find transition matrices from [v1, v2] to [u1, u2] and back again:a) Find the transition matrix V from [v1, v2] to the standard basis [e1, e2].b) Find the transition matrix U from [u1, u2] to the standard basis [e1, e2].c) Find the transition matrix from [v1, v2] to [u1, u2].d) Find the transition matrix from [u1, u2] to [v1, v2].e) Use your result from part c) to find the coordinates of w = v1 − v2 with respect to the basis [u1, u2]. (Note that [w][v1,v2] = (1, −1)T.)

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 18CM

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In the vector space R² consider the two bases given by [u₁, u₂] with u₁ = (1, 2)^T, u₂ = (2, 5)^T and [v₁, v₂] with v₁ = (3, 2)^T, v₂ = (4, 3)^T (both with respect to the standard basis).

Follow these steps to find transition matrices from [v₁, v₂] to [u₁, u₂] and back again:

a) Find the transition matrix V from [v₁, v₂] to the standard basis [e₁, e₂].
b) Find the transition matrix U from [u₁, u₂] to the standard basis [e₁, e₂].
c) Find the transition matrix from [v₁, v₂] to [u₁, u₂].
d) Find the transition matrix from [u₁, u₂] to [v₁, v₂].
e) Use your result from part c) to find the coordinates of w = v₁ − v₂ with respect to the basis [u₁, u₂]. (Note that [w]_[v1,v2] = (1, −1)^T.)

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