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1. Let (i) Dr (ii) Fi

1. Let (i) Dr (ii) Fi

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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Figure 4.2 for reference
## Problem Description 1. **Let** $$ A = \begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 0 & 1 \end{bmatrix}. $$ (i) Draw Figure 4.2 to show each subspace correctly for **A**. (ii) Find the pieces \( x_r \) and \( x_n \) and draw Figure 4.3 properly if \( x = (1, 2) \). ### Instructions - **Figure 4.2**: Illustrate each subspace for the matrix **A**. - **Figure 4.3**: Determine and illustrate the components \( x_r \) and \( x_n \) when \( x = (1, 2) \). ### Explanation - **Matrix Representation**: Matrix **A** is a 3x2 matrix, which can be used to describe transformations or systems of linear equations. - **Subspaces**: Subspaces related to matrix **A** might include its column space and null space. These geometric representations help in understanding the solutions and properties of the matrix. - **Component Analysis**: The vector \( x = (1, 2) \) can be decomposed into components relative to the column space (represented by \( x_r \)) and the null space (represented by \( x_n \)) of matrix **A**.
Transcribed Image Text:## Problem Description 1. **Let** $$ A = \begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 0 & 1 \end{bmatrix}. $$ (i) Draw Figure 4.2 to show each subspace correctly for **A**. (ii) Find the pieces \( x_r \) and \( x_n \) and draw Figure 4.3 properly if \( x = (1, 2) \). ### Instructions - **Figure 4.2**: Illustrate each subspace for the matrix **A**. - **Figure 4.3**: Determine and illustrate the components \( x_r \) and \( x_n \) when \( x = (1, 2) \). ### Explanation - **Matrix Representation**: Matrix **A** is a 3x2 matrix, which can be used to describe transformations or systems of linear equations. - **Subspaces**: Subspaces related to matrix **A** might include its column space and null space. These geometric representations help in understanding the solutions and properties of the matrix. - **Component Analysis**: The vector \( x = (1, 2) \) can be decomposed into components relative to the column space (represented by \( x_r \)) and the null space (represented by \( x_n \)) of matrix **A**.
**Orthogonality of the Four Subspaces** The diagram illustrates the four fundamental subspaces associated with a matrix \( A \) and highlights their orthogonality relationships. - **Row Space of \( A \):** - This is represented by a square on the top left. - The dimension is \( r \). - Arrows indicate a 90° angle to the Nullspace of \( A \). - **Nullspace of \( A \):** - This is shown as a square below the Row Space, also forming a 90° angle. - The dimension is \( n - r \). - **Column Space of \( A \):** - Displayed as a square on the top right. - The dimension is \( r \). - There is a 90° relationship with the Nullspace of \( A^T \). - **Nullspace of \( A^T \):** - Illustrated as a square below the Column Space. - The dimension is \( m - r \). **Figure Explanation:** - The figure (Figure 4.2) presents two pairs of orthogonal subspaces. These dimensions add up to \( n \) and \( m \), respectively. The diagram emphasizes two subspaces in \( \mathbb{R}^n \) and two subspaces in \( \mathbb{R}^m \). **Key Concepts:** - The orthogonality between these spaces is indicated by the 90° angles. - These relationships are central in understanding the fundamental theorem of linear algebra. This visual representation helps in grasping the orthogonal decomposition within different dimensions for matrices.
Transcribed Image Text:**Orthogonality of the Four Subspaces** The diagram illustrates the four fundamental subspaces associated with a matrix \( A \) and highlights their orthogonality relationships. - **Row Space of \( A \):** - This is represented by a square on the top left. - The dimension is \( r \). - Arrows indicate a 90° angle to the Nullspace of \( A \). - **Nullspace of \( A \):** - This is shown as a square below the Row Space, also forming a 90° angle. - The dimension is \( n - r \). - **Column Space of \( A \):** - Displayed as a square on the top right. - The dimension is \( r \). - There is a 90° relationship with the Nullspace of \( A^T \). - **Nullspace of \( A^T \):** - Illustrated as a square below the Column Space. - The dimension is \( m - r \). **Figure Explanation:** - The figure (Figure 4.2) presents two pairs of orthogonal subspaces. These dimensions add up to \( n \) and \( m \), respectively. The diagram emphasizes two subspaces in \( \mathbb{R}^n \) and two subspaces in \( \mathbb{R}^m \). **Key Concepts:** - The orthogonality between these spaces is indicated by the 90° angles. - These relationships are central in understanding the fundamental theorem of linear algebra. This visual representation helps in grasping the orthogonal decomposition within different dimensions for matrices.
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