please answer c thanks ### Change of Basis MatricesThis section provides the matrices used to convert between various custom coordinate systems and the standard coordinate system.#### a. From Custom \( \mathcal{B} \)-Coordinates to Standard \( \mathcal{S} \)-CoordinatesThe change of basis matrix from custom \( \mathcal{B} \)-coordinates to standard \( \mathcal{S} \)-coordinates is given by:\[[id]_{\mathcal{B}}^{\mathcal{S}} = \begin{bmatrix} 1 & -1 \\ -1 & -2 \end{bmatrix}\]#### b. From Custom \( \mathcal{C} \)-Coordinates to Standard \( \mathcal{S} \)-CoordinatesThe change of basis matrix from custom \( \mathcal{C} \)-coordinates to standard \( \mathcal{S} \)-coordinates is:\[[id]_{\mathcal{C}}^{\mathcal{S}} = \begin{bmatrix} -2 & -1 \\ -2 & 3 \end{bmatrix}\]#### c. From \( \mathcal{B} \)-Coordinates to \( \mathcal{C} \)-CoordinatesThe change of basis matrix from \( \mathcal{B} \)-coordinates to \( \mathcal{C} \)-coordinates is:\[[id]_{\mathcal{B}}^{\mathcal{C}} = \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix}\]#### d. From \( \mathcal{C} \)-Coordinates to \( \mathcal{B} \)-CoordinatesThe change of basis matrix from \( \mathcal{C} \)-coordinates to \( \mathcal{B} \)-coordinates is:\[[id]_{\mathcal{C}}^{\mathcal{B}} = \begin{bmatrix} -2/3 & -5/3 \\ 4/3 & -2/3 \end{bmatrix}\] ### Explanation of MatricesEach matrix represents a linear transformation that changes the representation of vectors from one coordinate system to another. The elements in these matrices are coefficients that define this transformation.- **Rows and Columns**: Each column of the matrix represents the transformation of one of the basis vectors.- **Values** The standard basis \( \mathcal{S} = \{\mathbf{e}_1, \mathbf{e}_2\} \) and two custom bases \( \mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2\} \) and \( \mathcal{C} = \{\mathbf{c}_1, \mathbf{c}_2\} \) for \( \mathbb{R}^2 \) are shown in the figures below.### Top Left Graph- **Description:** The graph shows the standard basis \( \mathcal{S} = \{\mathbf{e}_1, \mathbf{e}_2\} \).- **Vectors:** - \( \mathbf{e}_1\) in red, pointing along the positive x-axis. - \( \mathbf{e}_2\) in red, pointing along the positive y-axis. - \( \mathbf{b}_1\) in blue, forms a vector in the positive quadrant. - \( \mathbf{b}_2\) in blue, forms a vector in the negative quadrant.- **Label:** \([id]_{\mathcal{S}_B}\)### Top Right Graph- **Description:** The graph shows the standard basis \( \mathcal{S} = \{\mathbf{e}_1, \mathbf{e}_2\} \) overlaid with custom basis \( \mathcal{C} = \{\mathbf{c}_1, \mathbf{c}_2\} \).- **Vectors:** - \( \mathbf{e}_1\) in red, original position along x-axis. - \( \mathbf{e}_2\) in red, original position along y-axis. - \( \mathbf{c}_1\) in blue, points into the third quadrant. - \( \mathbf{c}_2\) in blue, points into the second quadrant.- **Label:** \([id]_{\mathcal{S}_S}\)### Bottom Left Graph- **Description:** The graph displays the custom basis \( \mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2\} \).- **Vectors:** - \( \mathbf{b}_1\) in blue, pointing into the fourth quadrant. - \( \mathbf{

The standard basis S = {ej , e2} and two custom bases B = {b1, b2} and C = {c1, c2} for R? are shown in the figures below. Standard basis S = {e1,e2} Standard basis S = {e1, e2} ly y c2 3 3 2 2 e2 e2 1 [id e1 e1 -1 b1 -2 -2 b2 c1 -3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 [id t [id ↑ c2 [id b1 b2 c1 Custom basis B = {b1, b2} Custom basis C = {c1, c2}