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Problem. Recall that D = {1, t, t²,..., t"} is a basis for P. and T₂(x) = [x]D denotes the D-coordinate mapping. Let T: P₂ → Pn denote the derivative (so, T(p) = p'). Both the domain and the codomain of the linear mapping F = T₂oToTp¹ are equal to R"+¹ and the mapping acts as follows: x → F(x) = T(T(T¹(x))). Find F(e;) for i = 1,..., n + 1. Find the standard matrix of this transformation when n = = 5.

Problem. Recall that D = {1, t, t²,..., t"} is a basis for P. and T₂(x) = [x]D denotes the D-coordinate mapping. Let T: P₂ → Pn denote the derivative (so, T(p) = p'). Both the domain and the codomain of the linear mapping F = T₂oToTp¹ are equal to R"+¹ and the mapping acts as follows: x → F(x) = T(T(T¹(x))). Find F(e;) for i = 1,..., n + 1. Find the standard matrix of this transformation when n = = 5.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 18EQ
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Problem. Recall that D = {1, t, t²,..., t} is a basis for P₁, and T₁(x) = [x]D denotes the D-coordinate mapping. Let T: Pn → Pn denote the derivative (so, T(p) = p'). Both the domain and the codomain of the linear mapping F = T₂°T° T₂¹ and the mapping acts as follows: x → F(x) = T₂(T(T₂¹(x))). Find F(e;) for i = 1,..., n + 1. Find the standard matrix of this transformation when n = 5. n+1 are equal to R"
Transcribed Image Text:Problem. Recall that D = {1, t, t²,..., t} is a basis for P₁, and T₁(x) = [x]D denotes the D-coordinate mapping. Let T: Pn → Pn denote the derivative (so, T(p) = p'). Both the domain and the codomain of the linear mapping F = T₂°T° T₂¹ and the mapping acts as follows: x → F(x) = T₂(T(T₂¹(x))). Find F(e;) for i = 1,..., n + 1. Find the standard matrix of this transformation when n = 5. n+1 are equal to R"
Solution. We need to check and see where the mapping sends the standard Coordinate vectors e₁, ..., en 1. Well: F(е₁) = T(T(T₂¹(е₁))) F(е₂) = TƊ(T(T₂¹(ẹ₂))) = T(T(⋆)) = T₂(*) = * = T(T(⋆)) = Td(*) = ★ F(е3) = TD(T(T₂¹(е3))) = T(T(⋆)) = Tp(*) = ★ F(е₁) = T₂(T(T₂¹(ẹ4))) = T₂(T(⋆)) = T₂(*) = ² F(en 1)=TD(T(Tp*(e» «ı)))=Tz(T(*))=Tz(*) = *. Replace every instance of above with the correct quantity. Delete this text. When n = 5, we obtain the matrix Typeset the matrix here. Delete this text.
Transcribed Image Text:Solution. We need to check and see where the mapping sends the standard Coordinate vectors e₁, ..., en 1. Well: F(е₁) = T(T(T₂¹(е₁))) F(е₂) = TƊ(T(T₂¹(ẹ₂))) = T(T(⋆)) = T₂(*) = * = T(T(⋆)) = Td(*) = ★ F(е3) = TD(T(T₂¹(е3))) = T(T(⋆)) = Tp(*) = ★ F(е₁) = T₂(T(T₂¹(ẹ4))) = T₂(T(⋆)) = T₂(*) = ² F(en 1)=TD(T(Tp*(e» «ı)))=Tz(T(*))=Tz(*) = *. Replace every instance of above with the correct quantity. Delete this text. When n = 5, we obtain the matrix Typeset the matrix here. Delete this text.
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