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)

Name: ### Linear Algebra Basis and Derivative Transformation#### Problem St
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Description: ### Linear Algebra Basis and Derivative Transformation#### Problem StatementConsider 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

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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Question

### 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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