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Consider the subspace spanned by B = {cos(x), sin(x), x cos(x), x sin(x)} inside the space of continuous functions. B is in fact a basis for its span. The derivative is a linear transformation on this span, and it acts as follows: D(cos(x)) = – sin(x) D(sin(x)) = cos(x) D(x cos(x)) = cos(x) – x sin(x) D(r sin(x)) = sin(x) + x cos(x) Find the matrix representing D in this basis. Use this to compute D(3 cos(x) – 2 sin(x) + x cos(x) – 2x sin(x)) without computing any derivatives. (show the matrix and column vector you use to do this, and the resulting column vector before converting back to functions)

Consider the subspace spanned by B = {cos(x), sin(x), x cos(x), x sin(x)} inside the space of continuous functions. B is in fact a basis for its span. The derivative is a linear transformation on this span, and it acts as follows: D(cos(x)) = – sin(x) D(sin(x)) = cos(x) D(x cos(x)) = cos(x) – x sin(x) D(r sin(x)) = sin(x) + x cos(x) Find the matrix representing D in this basis. Use this to compute D(3 cos(x) – 2 sin(x) + x cos(x) – 2x sin(x)) without computing any derivatives. (show the matrix and column vector you use to do this, and the resulting column vector before converting back to functions)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Linear Algebra Basis and Derivative Transformation #### Problem Statement Consider the subspace spanned by \( B = \{\cos(x), \sin(x), x \cos(x), x \sin(x)\} \) inside the space of continuous functions. \( B \) is in fact a basis for its span. The derivative is a linear transformation on this span, and it acts as follows: \[ D(\cos(x)) = -\sin(x) \] \[ D(\sin(x)) = \cos(x) \] \[ D(x \cos(x)) = \cos(x) - x \sin(x) \] \[ D(x \sin(x)) = \sin(x) + x \cos(x) \] #### Task: Find the matrix representing \( D \) in this basis. Use this to compute: \[ D(3 \cos(x) - 2 \sin(x) + x \cos(x) - 2x \sin(x)) \] without computing any derivatives. Show the matrix and column vector you use to do this, and the resulting column vector before converting back to functions. #### Solution: 1. **Derivatives in Terms of Basis:** - \( D(\cos(x)) = -\sin(x) \rightarrow -\sin(x) \) - \( D(\sin(x)) = \cos(x) \rightarrow \cos(x) \) - \( D(x \cos(x)) = \cos(x) - x \sin(x) \rightarrow \cos(x) - x \sin(x) \) - \( D(x \sin(x)) = \sin(x) + x \cos(x) \rightarrow \sin(x) + x \cos(x) \) 2. **Express Derivatives as Linear Combinations of the Basis:** - \( D(\cos(x)) = -\sin(x) \rightarrow [0,-1,0,0] \) - \( D(\sin(x)) = \cos(x) \rightarrow [1,0,0,0] \) - \( D(x \cos(x)) = \cos(x) - x \sin(x) \rightarrow [1,0,0,-1] \) - \( D(x \sin(x)) = \sin(x) + x \cos(x) \rightarrow [0,1,1,0] \) 3. **Matrix Representation:** The matrix \( [D] \) that
Transcribed Image Text:### Linear Algebra Basis and Derivative Transformation #### Problem Statement Consider the subspace spanned by \( B = \{\cos(x), \sin(x), x \cos(x), x \sin(x)\} \) inside the space of continuous functions. \( B \) is in fact a basis for its span. The derivative is a linear transformation on this span, and it acts as follows: \[ D(\cos(x)) = -\sin(x) \] \[ D(\sin(x)) = \cos(x) \] \[ D(x \cos(x)) = \cos(x) - x \sin(x) \] \[ D(x \sin(x)) = \sin(x) + x \cos(x) \] #### Task: Find the matrix representing \( D \) in this basis. Use this to compute: \[ D(3 \cos(x) - 2 \sin(x) + x \cos(x) - 2x \sin(x)) \] without computing any derivatives. Show the matrix and column vector you use to do this, and the resulting column vector before converting back to functions. #### Solution: 1. **Derivatives in Terms of Basis:** - \( D(\cos(x)) = -\sin(x) \rightarrow -\sin(x) \) - \( D(\sin(x)) = \cos(x) \rightarrow \cos(x) \) - \( D(x \cos(x)) = \cos(x) - x \sin(x) \rightarrow \cos(x) - x \sin(x) \) - \( D(x \sin(x)) = \sin(x) + x \cos(x) \rightarrow \sin(x) + x \cos(x) \) 2. **Express Derivatives as Linear Combinations of the Basis:** - \( D(\cos(x)) = -\sin(x) \rightarrow [0,-1,0,0] \) - \( D(\sin(x)) = \cos(x) \rightarrow [1,0,0,0] \) - \( D(x \cos(x)) = \cos(x) - x \sin(x) \rightarrow [1,0,0,-1] \) - \( D(x \sin(x)) = \sin(x) + x \cos(x) \rightarrow [0,1,1,0] \) 3. **Matrix Representation:** The matrix \( [D] \) that
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