Give [M], where M -3 8 9].
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Problem 1RQ
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![**Matrix Expression for Educational Purposes**
Consider the expression for the matrix \([M]_{\mathcal{E}}\), where the matrix \(M\) is given by:
\[
M = \begin{bmatrix}
-3 & 0 \\
8 & -4
\end{bmatrix}
\]
This matrix is a 2x2 matrix, with elements as follows:
- The element in the first row, first column is \(-3\).
- The element in the first row, second column is \(0\).
- The element in the second row, first column is \(8\).
- The element in the second row, second column is \(-4\).
The instruction is to give \([M]_{\mathcal{E}}\), which may refer to expressing the matrix \(M\) relative to a certain basis \(\mathcal{E}\) or performing a transformation related to this basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e470dde-d985-471c-b0b3-08e4bb7380ab%2F6f7bd549-e64e-45fb-aa07-9ee9fe5d117b%2F8wou2rl_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Expression for Educational Purposes**
Consider the expression for the matrix \([M]_{\mathcal{E}}\), where the matrix \(M\) is given by:
\[
M = \begin{bmatrix}
-3 & 0 \\
8 & -4
\end{bmatrix}
\]
This matrix is a 2x2 matrix, with elements as follows:
- The element in the first row, first column is \(-3\).
- The element in the first row, second column is \(0\).
- The element in the second row, first column is \(8\).
- The element in the second row, second column is \(-4\).
The instruction is to give \([M]_{\mathcal{E}}\), which may refer to expressing the matrix \(M\) relative to a certain basis \(\mathcal{E}\) or performing a transformation related to this basis.
![### Understanding Matrix Subspaces
**Given the Subspace:**
\[
H = \left\{ \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \middle| \ a, b, c \in \mathbb{R} \right\}
\]
This notation defines \( H \) as the space consisting of 2x2 lower triangular matrices where each matrix has elements \( a, b, \) and \( c \) that are real numbers.
**Nature of \( H \):**
- \( H \) is identified as the space of lower triangular 2x2 matrices. A matrix is lower triangular if all elements above the main diagonal are zero.
**Basis of the Subspace:**
The set of matrices below is established as a basis for \( H \):
\[
\mathcal{E} = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}
\]
- **Explanation of the Basis:**
- The basis consists of matrices that span the subspace \( H \).
- Each matrix represents a unique component of the lower triangular form.
- The first matrix, \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), contributes to the variation of the \(a\) component,
- The second, \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\), influences the \(b\) component,
- Lastly, \(\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\), represents the \(c\) component.
These matrices enable any matrix in \( H \) to be expressed as a linear combination of the basis elements, demonstrating that they indeed form a basis of \( H \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e470dde-d985-471c-b0b3-08e4bb7380ab%2F6f7bd549-e64e-45fb-aa07-9ee9fe5d117b%2F3n4i2sf_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Matrix Subspaces
**Given the Subspace:**
\[
H = \left\{ \begin{bmatrix} a & 0 \\ b & c \end{bmatrix} \middle| \ a, b, c \in \mathbb{R} \right\}
\]
This notation defines \( H \) as the space consisting of 2x2 lower triangular matrices where each matrix has elements \( a, b, \) and \( c \) that are real numbers.
**Nature of \( H \):**
- \( H \) is identified as the space of lower triangular 2x2 matrices. A matrix is lower triangular if all elements above the main diagonal are zero.
**Basis of the Subspace:**
The set of matrices below is established as a basis for \( H \):
\[
\mathcal{E} = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}
\]
- **Explanation of the Basis:**
- The basis consists of matrices that span the subspace \( H \).
- Each matrix represents a unique component of the lower triangular form.
- The first matrix, \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), contributes to the variation of the \(a\) component,
- The second, \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}\), influences the \(b\) component,
- Lastly, \(\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\), represents the \(c\) component.
These matrices enable any matrix in \( H \) to be expressed as a linear combination of the basis elements, demonstrating that they indeed form a basis of \( H \).
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