help Let \( V \) be the vector space of functions spanned by \[\mathcal{B} = \{\mathbf{b}_1 = 3 \sin x + 4 \cos x, \, \mathbf{b}_2 = 8 \sin x + 9 \cos x\}\] where \( x \neq \frac{n\pi}{2}, \, n \in \mathbb{Z} \).\[\mathcal{C} = \{\mathbf{c}_1 = \sin x, \, \mathbf{c}_2 = \cos x\}\] where \( x \neq \frac{n\pi}{2}, \, n \in \mathbb{Z} \) is also a basis for \( V \). Find the change of coordinates matrix \( P_{\mathcal{C} \leftarrow \mathcal{B}} \).\[P_{\mathcal{C} \leftarrow \mathcal{B}} = \begin{bmatrix}\text{Ex: 7} & \\& \end{bmatrix}\]The coordinates of a function \( f(x) \) relative to the basis \( \mathcal{B} \) are \[[f(x)]_{\mathcal{B}} = \begin{bmatrix}5 \\6 \end{bmatrix}\]Find the coordinates of \( f(x) \) relative to \( \mathcal{C} \) and find \( f(x) \).\[[f(x)]_{\mathcal{C}} = \begin{bmatrix}\text{Ex: 7} \\\end{bmatrix}\]\[f(x) = \text{Ex: 7} \, \sin x + \, \cos x\]

Name: help Let \( V \) be the vector space of functions spanned by \[\mathca
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: help Let \( V \) be the vector space of functions spanned by \[\mathcal{B} = \{\mathbf{b}_1 = 3 \sin x + 4 \cos x, \, \mathbf{b}_2 = 8 \sin x + 9 \cos x\}\] where \( x \neq \frac{n\pi}{2}, \, n \in \mathbb{Z} \).\[\mathcal{C} = \{\mathbf{c}_1 = \sin

The basis β is b1=3sinx+4cosx, b2=8sinx+9cosx and basis C is sinx, cosx. The change of coordinate matrix is obtained as PC←β =b1Cb2C=3849 Given: fxβ =56 Then, the function is fx=53sinx+4cosx+68sinx+9cosx=15sinx+20cosx+48sinx+54cosx=63sinx+74cosx fxC=63sinx+74cosxC=6374 Alternate way, fxC=PC←β ·fxβ=3849·56=6374 Hence, fx=63sinx+74cosx