Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### 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)](https://content.bartleby.com/qna-images/question/31b13744-2a7c-4c52-8a63-0a640293092a/1a37a3f0-7deb-48e6-a0f8-ff4ecd447bba/jcq9vn_processed.jpeg)
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)
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