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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.

Advanced Engineering Mathematics
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: 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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