-2 5 5 -38 3 -2 -6 40 A = and b = -5 -3 2 -22 -17 -25 -1 -14 Define the linear transformation T: R³ → R² by T(x) = = Ax. Find a vector whose image under T is b. x = Is the vector a unique? choose LO
-2 5 5 -38 3 -2 -6 40 A = and b = -5 -3 2 -22 -17 -25 -1 -14 Define the linear transformation T: R³ → R² by T(x) = = Ax. Find a vector whose image under T is b. x = Is the vector a unique? choose LO
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![## Linear Transformations and Vector Solutions
Consider the matrix \( A \) and the vector \( \vec{b} \) given by:
\[
A = \begin{bmatrix}
-2 & 5 & 5 \\
3 & -2 & -6 \\
-5 & -3 & 2 \\
-17 & -25 & -1
\end{bmatrix}
\quad \text{and} \quad
\vec{b} = \begin{bmatrix}
-38 \\
40 \\
-22 \\
-14
\end{bmatrix}.
\]
Define the linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^4 \) by \( T(\vec{x}) = A\vec{x} \). We need to find a vector \( \vec{x} \) whose image under \( T \) is \( \vec{b} \).
\[
\vec{x} = \begin{bmatrix}
\boxed{\phantom{x}} \\
\boxed{\phantom{x}} \\
\boxed{\phantom{x}}
\end{bmatrix}.
\]
Is the vector \( \vec{x} \) unique?
\[ \text{choose} \]
### Task Explanation
1. **Matrix \( A \)**: This is a \( 4 \times 3 \) matrix, meaning it represents a transformation from \(\mathbb{R}^3\) to \(\mathbb{R}^4\).
2. **Vector \( \vec{b} \)**: This is a \( 4 \times 1 \) vector (or a column vector) to which we want to map the output of the transformation.
3. **Linear Transformation \( T \)**: The transformation maps a vector \(\vec{x}\) from the 3-dimensional space to a 4-dimensional space using matrix multiplication with \( A \).
### Objective
To find a vector \( \vec{x} \) in \(\mathbb{R}^3\) such that when we apply the linear transformation \( T \) to this vector, we obtain \( \vec{b} \). Mathematically, we need to solve the equation:
\[ A\vec{x} = \vec{b} \]
We will also determine if the solution vector \( \vec{x} \) is unique](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad281aeb-79af-4eda-b9b4-346ba477171b%2F7467a195-b6f9-49ef-8a07-524ca2c5b0e0%2Fv80dws_processed.png&w=3840&q=75)
Transcribed Image Text:## Linear Transformations and Vector Solutions
Consider the matrix \( A \) and the vector \( \vec{b} \) given by:
\[
A = \begin{bmatrix}
-2 & 5 & 5 \\
3 & -2 & -6 \\
-5 & -3 & 2 \\
-17 & -25 & -1
\end{bmatrix}
\quad \text{and} \quad
\vec{b} = \begin{bmatrix}
-38 \\
40 \\
-22 \\
-14
\end{bmatrix}.
\]
Define the linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^4 \) by \( T(\vec{x}) = A\vec{x} \). We need to find a vector \( \vec{x} \) whose image under \( T \) is \( \vec{b} \).
\[
\vec{x} = \begin{bmatrix}
\boxed{\phantom{x}} \\
\boxed{\phantom{x}} \\
\boxed{\phantom{x}}
\end{bmatrix}.
\]
Is the vector \( \vec{x} \) unique?
\[ \text{choose} \]
### Task Explanation
1. **Matrix \( A \)**: This is a \( 4 \times 3 \) matrix, meaning it represents a transformation from \(\mathbb{R}^3\) to \(\mathbb{R}^4\).
2. **Vector \( \vec{b} \)**: This is a \( 4 \times 1 \) vector (or a column vector) to which we want to map the output of the transformation.
3. **Linear Transformation \( T \)**: The transformation maps a vector \(\vec{x}\) from the 3-dimensional space to a 4-dimensional space using matrix multiplication with \( A \).
### Objective
To find a vector \( \vec{x} \) in \(\mathbb{R}^3\) such that when we apply the linear transformation \( T \) to this vector, we obtain \( \vec{b} \). Mathematically, we need to solve the equation:
\[ A\vec{x} = \vec{b} \]
We will also determine if the solution vector \( \vec{x} \) is unique
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