### 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 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

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