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The matrix that implements the mapping T (₁, 2) = (-3x1 +4x2,0, 7x2 - 5x1,1 + 2x₂) is -3071 4 052 0 O O C O 51 199 351 472 -3 4 7 -5 1 2 -3 4 0 0 7 -5 1 2 < Previous

The matrix that implements the mapping T (₁, 2) = (-3x1 +4x2,0, 7x2 - 5x1,1 + 2x₂) is -3071 4 052 0 O O C O 51 199 351 472 -3 4 7 -5 1 2 -3 4 0 0 7 -5 1 2 < Previous

Advanced Engineering Mathematics
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 10 **Which of the following statements is true?** 1. ☐ If \( A \) is a \( 4 \times 3 \) matrix, then the transformation \( \mathbf{x} \rightarrow A\mathbf{x} \) cannot be one to one. 2. ☐ When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. 3. ☐ A mapping \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is one to one if each vector in \( \mathbb{R}^m \) maps onto a unique vector in \( \mathbb{R}^n \) . 4. ☐ The columns of the standard matrix for a linear transformation \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) are the images of the columns of the \( n \times n \) identity matrix. 5. ☐ Not every linear transformation \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a matrix transformation. **Explanation:** - **Option 1:** This statement suggests that a linear transformation induced by a \( 4 \times 3 \) matrix cannot be one to one. - **Option 2:** This indicates that the combined effect of two sequential linear transformations might not always result in a linear transformation, which is typically a misunderstanding of linear transformations' properties. - **Option 3:** This statement talks about injectivity (one-to-one property) of the mapping, suggesting a vector in the codomain uniquely maps back to a vector in the domain. - **Option 4:** This option describes the standard matrix representation of a linear transformation and indicates that the columns of this matrix correspond to the images of the identity matrix columns. - **Option 5:** This states that not every linear transformation can be expressed as a matrix transformation. **Navigation:** - **Previous Button:** Allows the user to navigate to the previous question or section. **Status Check:** - **No new data to save. Last checked at 5:03am** This image contains a multiple-choice question related to linear algebra, particularly addressing properties and characterizations of linear transformations and matrices. The user is expected to select the statement which is true based on their understanding of the subject
Transcribed Image Text:### Question 10 **Which of the following statements is true?** 1. ☐ If \( A \) is a \( 4 \times 3 \) matrix, then the transformation \( \mathbf{x} \rightarrow A\mathbf{x} \) cannot be one to one. 2. ☐ When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. 3. ☐ A mapping \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is one to one if each vector in \( \mathbb{R}^m \) maps onto a unique vector in \( \mathbb{R}^n \) . 4. ☐ The columns of the standard matrix for a linear transformation \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) are the images of the columns of the \( n \times n \) identity matrix. 5. ☐ Not every linear transformation \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a matrix transformation. **Explanation:** - **Option 1:** This statement suggests that a linear transformation induced by a \( 4 \times 3 \) matrix cannot be one to one. - **Option 2:** This indicates that the combined effect of two sequential linear transformations might not always result in a linear transformation, which is typically a misunderstanding of linear transformations' properties. - **Option 3:** This statement talks about injectivity (one-to-one property) of the mapping, suggesting a vector in the codomain uniquely maps back to a vector in the domain. - **Option 4:** This option describes the standard matrix representation of a linear transformation and indicates that the columns of this matrix correspond to the images of the identity matrix columns. - **Option 5:** This states that not every linear transformation can be expressed as a matrix transformation. **Navigation:** - **Previous Button:** Allows the user to navigate to the previous question or section. **Status Check:** - **No new data to save. Last checked at 5:03am** This image contains a multiple-choice question related to linear algebra, particularly addressing properties and characterizations of linear transformations and matrices. The user is expected to select the statement which is true based on their understanding of the subject
### Question: The matrix that implements the mapping \( T(x_1, x_2) = (-3x_1 + 4x_2, 0, 7x_2 - 5x_1, x_1 + 2x_2) \) is: 1. \[ \begin{pmatrix} -3 & 0 & 7 & 1 \\ 4 & 0 & 5 & 2 \end{pmatrix} \] 2. \[ \begin{pmatrix} -3 & 4 \\ 0 & 0 \\ 5 & 7 \\ 1 & 2 \end{pmatrix} \] 3. \[ \begin{pmatrix} -3 & 4 \\ -5 & 7 \\ 1 & 2 \end{pmatrix} \] 4. \[ \begin{pmatrix} -3 & 4 \\ 7 & -5 \\ 1 & 2 \end{pmatrix} \] 5. \[ \begin{pmatrix} -3 & 4 \\ 0 & 0 \\ 7 & -5 \\ 1 & 2 \end{pmatrix} \]
Transcribed Image Text:### Question: The matrix that implements the mapping \( T(x_1, x_2) = (-3x_1 + 4x_2, 0, 7x_2 - 5x_1, x_1 + 2x_2) \) is: 1. \[ \begin{pmatrix} -3 & 0 & 7 & 1 \\ 4 & 0 & 5 & 2 \end{pmatrix} \] 2. \[ \begin{pmatrix} -3 & 4 \\ 0 & 0 \\ 5 & 7 \\ 1 & 2 \end{pmatrix} \] 3. \[ \begin{pmatrix} -3 & 4 \\ -5 & 7 \\ 1 & 2 \end{pmatrix} \] 4. \[ \begin{pmatrix} -3 & 4 \\ 7 & -5 \\ 1 & 2 \end{pmatrix} \] 5. \[ \begin{pmatrix} -3 & 4 \\ 0 & 0 \\ 7 & -5 \\ 1 & 2 \end{pmatrix} \]
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