**Problem 34: Linear Transformation Example**- **Objective**: Provide an example of a linear transformation whose kernel is the line spanned by the vector $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$ in $\mathbb{R}^3$.- **Exercise**: How would you solve this using matrix calculations such as $A \vec{x} = \vec{0}$?To find such a linear transformation, consider a matrix $A$ that when multiplied by the vector $\vec{x} = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$, yields the zero vector $\vec{0}$. This indicates that the vector is part of the kernel of the transformation. The process involves setting up a system of equations representing linear combinations of the rows of $A$ that result in the specified zero output for the given vector, ensuring all vectors in the kernel are scalar multiples of $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$.

Answered: 34. Give an example of a linear…

34. Give an example of a linear transformation whose ker- nel is the line spanned by in R³. How would you oblue this usineg Matrix caletafion such as ン

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

**Problem 34: Linear Transformation Example**

- **Objective**: Provide an example of a linear transformation whose kernel is the line spanned by the vector $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$ in $\mathbb{R}^3$.

- **Exercise**: How would you solve this using matrix calculations such as $A \vec{x} = \vec{0}$?

To find such a linear transformation, consider a matrix $A$ that when multiplied by the vector $\vec{x} = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$, yields the zero vector $\vec{0}$. This indicates that the vector is part of the kernel of the transformation. The process involves setting up a system of equations representing linear combinations of the rows of $A$ that result in the specified zero output for the given vector, ensuring all vectors in the kernel are scalar multiples of $\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$.

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