3. The linear operator T: R³ R³ is defined by T(x1, x2, x3) = (W1₁, W2, W3), where w₁ = 2x₁ + 4x2 + x3; W2 = 9x2 + 2x3; W3 = 2x1 - 8x2 - 2x3. Which of the following is correct. (a) T is not one to one. (b) T is one to one but the standard matrix for T-1 does not exist. 0 (c) T is one to one and its standard matrix for T-¹ is 1 -4 -3 3 3 (d) T is one to one and its standard matrix for T-1 is 0 1 -4 -3 1 2 6 3 (e) None of these 13 23 3 2/31 2²/12 3 162/3 3
3. The linear operator T: R³ R³ is defined by T(x1, x2, x3) = (W1₁, W2, W3), where w₁ = 2x₁ + 4x2 + x3; W2 = 9x2 + 2x3; W3 = 2x1 - 8x2 - 2x3. Which of the following is correct. (a) T is not one to one. (b) T is one to one but the standard matrix for T-1 does not exist. 0 (c) T is one to one and its standard matrix for T-¹ is 1 -4 -3 3 3 (d) T is one to one and its standard matrix for T-1 is 0 1 -4 -3 1 2 6 3 (e) None of these 13 23 3 2/31 2²/12 3 162/3 3
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
![### Linear Operator: Evaluation and Properties
Consider the linear operator \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by \( T(x_1, x_2, x_3) = (w_1, w_2, w_3) \), where:
\[ w_1 = 2x_1 + 4x_2 + x_3 \]
\[ w_2 = 9x_2 + 2x_3 \]
\[ w_3 = 2x_1 - 8x_2 - 2x_3 \]
**Question:** Which of the following statements is correct?
(a) \( T \) is not one to one.
(b) \( T \) is one to one.
(c) \( T \) is one to one but the standard matrix for \( T^{-1} \) does not exist.
(d) \( T \) is one to one and its standard matrix for \( T^{-1} \) is
\[ \begin{pmatrix}
\frac{1}{6} & 0 & \frac{1}{6} \\
0 & \frac{1}{3} & \frac{1}{3} \\
\frac{1}{12} & -\frac{1}{4} & -\frac{7}{12}
\end{pmatrix} \]
(e) None of these.
### Analysis of Options
#### Option (a)
- **Statement:** \( T \) is not one to one.
- **Analysis:** To verify this, we need to check the kernel of \( T \). If the kernel contains only the zero vector, \( T \) is one to one. Otherwise, it is not.
#### Option (b)
- **Statement:** \( T \) is one to one.
- **Analysis:** This implies that the transformation is injective. We can verify this by determining if the determinant of the transformation matrix is non-zero.
#### Option (c)
- **Statement:** \( T \) is one to one but the standard matrix for \( T^{-1} \) does not exist.
- **Analysis:** If \( T \) is one to one, the inverse exists. For the inverse matrix to exist and be valid, the standard matrix of \( T \) should be invertible.
#### Option (d)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31b13744-2a7c-4c52-8a63-0a640293092a%2F1a37a3f0-7deb-48e6-a0f8-ff4ecd447bba%2Fjcq9vn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Operator: Evaluation and Properties
Consider the linear operator \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by \( T(x_1, x_2, x_3) = (w_1, w_2, w_3) \), where:
\[ w_1 = 2x_1 + 4x_2 + x_3 \]
\[ w_2 = 9x_2 + 2x_3 \]
\[ w_3 = 2x_1 - 8x_2 - 2x_3 \]
**Question:** Which of the following statements is correct?
(a) \( T \) is not one to one.
(b) \( T \) is one to one.
(c) \( T \) is one to one but the standard matrix for \( T^{-1} \) does not exist.
(d) \( T \) is one to one and its standard matrix for \( T^{-1} \) is
\[ \begin{pmatrix}
\frac{1}{6} & 0 & \frac{1}{6} \\
0 & \frac{1}{3} & \frac{1}{3} \\
\frac{1}{12} & -\frac{1}{4} & -\frac{7}{12}
\end{pmatrix} \]
(e) None of these.
### Analysis of Options
#### Option (a)
- **Statement:** \( T \) is not one to one.
- **Analysis:** To verify this, we need to check the kernel of \( T \). If the kernel contains only the zero vector, \( T \) is one to one. Otherwise, it is not.
#### Option (b)
- **Statement:** \( T \) is one to one.
- **Analysis:** This implies that the transformation is injective. We can verify this by determining if the determinant of the transformation matrix is non-zero.
#### Option (c)
- **Statement:** \( T \) is one to one but the standard matrix for \( T^{-1} \) does not exist.
- **Analysis:** If \( T \) is one to one, the inverse exists. For the inverse matrix to exist and be valid, the standard matrix of \( T \) should be invertible.
#### Option (d)
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 4 steps with 4 images
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,
