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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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one question with all subparts
![**Linear Algebra Problem Set: Linear Operator on \( \mathbb{R}^3 \)**
Suppose \( L \) is a linear operator on \( \mathbb{R}^3 \).
We also know:
\[
L \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -2 \\ 1 \\ 6 \end{pmatrix}, \quad
L \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}, \quad
L \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 6 \end{pmatrix}
\]
**Tasks:**
a) Find \( A \), the matrix representation for \( L \), with respect to the standard basis.
b) If \( \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \), find the "matrix rule" for \( L \) by computing \( A \vec{x} \).
c) Solve \( A \vec{x} = \vec{0} \).
d) Give a basis for the kernel of \( L \).
e) Find a spanning set for the range of \( L \).
f) Find a basis for the range of \( L \).
g) Use the answer in f) to find any specific element in the range of \( L \).
h) Could an inverse transformation \( L^{-1} \) exist? Use work to support the answer.
---
**Note:** Please provide clear, step-by-step calculations for each part to ensure a comprehensive understanding of the linear transformation and its properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F917b1102-0baf-48e9-8b4b-660a373cfe1a%2Fe7007c2d-7e84-48ba-ab4b-80e5c4fe9bbb%2Fraduzzt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Problem Set: Linear Operator on \( \mathbb{R}^3 \)**
Suppose \( L \) is a linear operator on \( \mathbb{R}^3 \).
We also know:
\[
L \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -2 \\ 1 \\ 6 \end{pmatrix}, \quad
L \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}, \quad
L \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 6 \end{pmatrix}
\]
**Tasks:**
a) Find \( A \), the matrix representation for \( L \), with respect to the standard basis.
b) If \( \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \), find the "matrix rule" for \( L \) by computing \( A \vec{x} \).
c) Solve \( A \vec{x} = \vec{0} \).
d) Give a basis for the kernel of \( L \).
e) Find a spanning set for the range of \( L \).
f) Find a basis for the range of \( L \).
g) Use the answer in f) to find any specific element in the range of \( L \).
h) Could an inverse transformation \( L^{-1} \) exist? Use work to support the answer.
---
**Note:** Please provide clear, step-by-step calculations for each part to ensure a comprehensive understanding of the linear transformation and its properties.
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