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}
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}
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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please answer c thanks
![### Change of Basis Matrices
This 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 \( \mathcal{S} \)-Coordinates
The 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 \( \mathcal{S} \)-Coordinates
The 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 \( \mathcal{C} \)-Coordinates
The 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 \( \mathcal{B} \)-Coordinates
The 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 Matrices
Each 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**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F415cd6f3-0f7d-47fa-9f1f-e0c2f10f61bd%2F564c544e-d846-4fdf-b03a-b2ef4fe037d8%2Fl371ywg_processed.png&w=3840&q=75)
Transcribed Image Text:### Change of Basis Matrices
This 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 \( \mathcal{S} \)-Coordinates
The 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 \( \mathcal{S} \)-Coordinates
The 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 \( \mathcal{C} \)-Coordinates
The 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 \( \mathcal{B} \)-Coordinates
The 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 Matrices
Each 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{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F415cd6f3-0f7d-47fa-9f1f-e0c2f10f61bd%2F564c544e-d846-4fdf-b03a-b2ef4fe037d8%2Fzaui5o9_processed.png&w=3840&q=75)
Transcribed Image Text: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{
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