) Let T : R² → R³ be a linear transformation such that 2 and T(( 1 (:) Is the vector u = (*) in R such that in the range of T, i.e., is there a vector 1 T X2 = u.
) Let T : R² → R³ be a linear transformation such that 2 and T(( 1 (:) Is the vector u = (*) in R such that in the range of T, i.e., is there a vector 1 T X2 = u.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let \( T: \mathbb{R}^2 \to \mathbb{R}^3 \) be a linear transformation such that
\[
T\left( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad T\left( \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}.
\]
Is the vector \( \mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \) in the range of \( T \), i.e., is there a vector \( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \) in \( \mathbb{R}^2 \) such that
\[
T\left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) = \mathbf{u}?
\]
Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87bd9bd0-40fd-4172-a50a-abb52eb6a8c1%2F4f646971-bf60-48cd-95f1-2d2035adba88%2F8q0iu2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( T: \mathbb{R}^2 \to \mathbb{R}^3 \) be a linear transformation such that
\[
T\left( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad T\left( \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}.
\]
Is the vector \( \mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \) in the range of \( T \), i.e., is there a vector \( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \) in \( \mathbb{R}^2 \) such that
\[
T\left( \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right) = \mathbf{u}?
\]
Justify your answer.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
