Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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I would appreciate your help in solving this problem using matrix notation alone. I am encountering challenges in finding a solution without relying on any other methods. Could you kindly provide a detailed, step-by-step explanation using only matrix notation to guide me to the final solution?
Furthermore, I have attached the question and answer for your reference. Can you demonstrate the matrix-based approach that leads to the solution?
Has to be the matrices way
![### Linear Transformation Problem
#### Problem Statement
Let \( T : \mathbb{R}^3 \to \mathbb{R}^2 \) be a linear transformation, and consider the following vectors:
\[
\mathbf{u} = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix},
\mathbf{v} = \begin{bmatrix}
2 \\
3 \\
1
\end{bmatrix},
\mathbf{w} = \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix},
\text{ and }
\mathbf{x} = \begin{bmatrix}
1 \\
2 \\
9
\end{bmatrix}.
\]
Given that:
\[
T(\mathbf{u}) = \begin{bmatrix}
1 \\
2
\end{bmatrix},
T(\mathbf{v}) = \begin{bmatrix}
2 \\
-1
\end{bmatrix},
\text{ and }
T(\mathbf{w}) = \begin{bmatrix}
2 \\
3
\end{bmatrix},
\]
compute \( T(\mathbf{x}) \).
### Explanation of Terms
- **Linear Transformation:** A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
- **Vectors:** Arrays of numbers that represent points or directions in space.
### Detailed Explanation:
You are provided with three known vectors in \(\mathbb{R}^3\) space and their corresponding transformations in \(\mathbb{R}^2\). You need to use this information to compute the transformation of the new vector \(\mathbf{x}\).
### Step-by-Step Solution:
1. **Expressing the new vector as a linear combination:**
Given vectors:
\[
\mathbf{u} = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix},
\mathbf{v} = \begin{bmatrix}
2 \\
3 \\
1
\end{bmatrix},
\mathbf{w} = \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}
\]
and transformation results:
\[](https://content.bartleby.com/qna-images/question/db2530dd-2d1e-49b0-a27c-5f0f04c951ed/5c7f8ca6-dc9d-4ef9-85de-8e2a94540664/p9p3hp_processed.png)
Transcribed Image Text:### Linear Transformation Problem
#### Problem Statement
Let \( T : \mathbb{R}^3 \to \mathbb{R}^2 \) be a linear transformation, and consider the following vectors:
\[
\mathbf{u} = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix},
\mathbf{v} = \begin{bmatrix}
2 \\
3 \\
1
\end{bmatrix},
\mathbf{w} = \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix},
\text{ and }
\mathbf{x} = \begin{bmatrix}
1 \\
2 \\
9
\end{bmatrix}.
\]
Given that:
\[
T(\mathbf{u}) = \begin{bmatrix}
1 \\
2
\end{bmatrix},
T(\mathbf{v}) = \begin{bmatrix}
2 \\
-1
\end{bmatrix},
\text{ and }
T(\mathbf{w}) = \begin{bmatrix}
2 \\
3
\end{bmatrix},
\]
compute \( T(\mathbf{x}) \).
### Explanation of Terms
- **Linear Transformation:** A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
- **Vectors:** Arrays of numbers that represent points or directions in space.
### Detailed Explanation:
You are provided with three known vectors in \(\mathbb{R}^3\) space and their corresponding transformations in \(\mathbb{R}^2\). You need to use this information to compute the transformation of the new vector \(\mathbf{x}\).
### Step-by-Step Solution:
1. **Expressing the new vector as a linear combination:**
Given vectors:
\[
\mathbf{u} = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix},
\mathbf{v} = \begin{bmatrix}
2 \\
3 \\
1
\end{bmatrix},
\mathbf{w} = \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}
\]
and transformation results:
\[
![## Problem 7.10
**Task:** Write **x** as a linear combination of **u**, **v**, **w**. Given that \( T(\mathbf{x}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix} \).
### Explanation
This problem involves expressing the vector **x** as a linear combination of the vectors **u**, **v**, and **w**. The transformation \( T \) applied to **x** results in a column vector \( \begin{bmatrix} 3 \\ 14 \end{bmatrix} \).
### Steps to solve:
1. **Understanding Linear Combination:**
- If **x** is written as a linear combination of **u**, **v**, and **w**, it means there exist scalars \( a \), \( b \), and \( c \) such that:
\[
\mathbf{x} = a\mathbf{u} + b\mathbf{v} + c\mathbf{w}
\]
2. **Applying the transformation \( T \):**
- The linear transformation \( T \) applied to **x** should equal the given vector:
\[
T(\mathbf{x}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix}
\]
3. **Setting up the equations:**
- Substitute the linear combination into the transformation equation:
\[
T(a\mathbf{u} + b\mathbf{v} + c\mathbf{w}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix}
\]
4. **Solving for scalars \( a \), \( b \), and \( c \):**
- This typically involves setting up a system of linear equations to find the values of \( a \), \( b \), and \( c \) that satisfy the equation.
### Diagrams:
Since the problem does not provide explicit vectors **u**, **v**, and **w**, the detailed solution would require those vectors. However, the general method would involve forming a matrix from **u**, **v**, and **w**, then solving the resulting system of equations.
### Example (Hypothetical Vectors):
If hypothetically, **u**, **v**, and **w** were given, the system of equations would be formed as follows:
Given \( \mathbf{u](https://content.bartleby.com/qna-images/question/db2530dd-2d1e-49b0-a27c-5f0f04c951ed/5c7f8ca6-dc9d-4ef9-85de-8e2a94540664/bg54uck_processed.png)
Transcribed Image Text:## Problem 7.10
**Task:** Write **x** as a linear combination of **u**, **v**, **w**. Given that \( T(\mathbf{x}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix} \).
### Explanation
This problem involves expressing the vector **x** as a linear combination of the vectors **u**, **v**, and **w**. The transformation \( T \) applied to **x** results in a column vector \( \begin{bmatrix} 3 \\ 14 \end{bmatrix} \).
### Steps to solve:
1. **Understanding Linear Combination:**
- If **x** is written as a linear combination of **u**, **v**, and **w**, it means there exist scalars \( a \), \( b \), and \( c \) such that:
\[
\mathbf{x} = a\mathbf{u} + b\mathbf{v} + c\mathbf{w}
\]
2. **Applying the transformation \( T \):**
- The linear transformation \( T \) applied to **x** should equal the given vector:
\[
T(\mathbf{x}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix}
\]
3. **Setting up the equations:**
- Substitute the linear combination into the transformation equation:
\[
T(a\mathbf{u} + b\mathbf{v} + c\mathbf{w}) = \begin{bmatrix} 3 \\ 14 \end{bmatrix}
\]
4. **Solving for scalars \( a \), \( b \), and \( c \):**
- This typically involves setting up a system of linear equations to find the values of \( a \), \( b \), and \( c \) that satisfy the equation.
### Diagrams:
Since the problem does not provide explicit vectors **u**, **v**, and **w**, the detailed solution would require those vectors. However, the general method would involve forming a matrix from **u**, **v**, and **w**, then solving the resulting system of equations.
### Example (Hypothetical Vectors):
If hypothetically, **u**, **v**, and **w** were given, the system of equations would be formed as follows:
Given \( \mathbf{u
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