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Question: Differentiation provides an interesting example of a linear transformation between spaces of different dimension. For concreteness, think of differentiation as a transformation from R4 to R³, where the elements of Rª are regarded as the coordinate vectors of elements of P3 (R) and the elements of R³ are regarded as the coordinate vectors of elements of P₂ (R), both with respect to the standard bases. Use Theorem to find the matrix of the "differentiation" transformation.

Question: Differentiation provides an interesting example of a linear transformation between spaces of different dimension. For concreteness, think of differentiation as a transformation from R4 to R³, where the elements of Rª are regarded as the coordinate vectors of elements of P3 (R) and the elements of R³ are regarded as the coordinate vectors of elements of P₂ (R), both with respect to the standard bases. Use Theorem to find the matrix of the "differentiation" transformation.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 21E
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Question: Differentiation provides an interesting example of a linear transformation between spaces of different dimension. For concreteness, think of differentiation as a transformation from R4 to R³, where the elements of Rª are regarded as the coordinate vectors of elements of P3 (R) and the elements of R³ are regarded as the coordinate vectors of elements of P₂ (R), both with respect to the standard bases. Use Theorem to find the matrix of the "differentiation" transformation. Theorem: The Standard Matrix of a Linear Transformation - Given a transformation T: R" to Rm T is linear if and only if T(v) Mv where Mij = T(ej), j = 1, 2, ..., n. M is called the standard matrix of T. =
Transcribed Image Text:Question: Differentiation provides an interesting example of a linear transformation between spaces of different dimension. For concreteness, think of differentiation as a transformation from R4 to R³, where the elements of Rª are regarded as the coordinate vectors of elements of P3 (R) and the elements of R³ are regarded as the coordinate vectors of elements of P₂ (R), both with respect to the standard bases. Use Theorem to find the matrix of the "differentiation" transformation. Theorem: The Standard Matrix of a Linear Transformation - Given a transformation T: R" to Rm T is linear if and only if T(v) Mv where Mij = T(ej), j = 1, 2, ..., n. M is called the standard matrix of T. =
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