B = [3 {[9][9]} 02 4 0 56
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
Is this a basis for M2×2? Can you explain why?
![**Matrix Set Representation:**
The set \( B \) is defined as:
\[
B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, \begin{bmatrix} 3 & 0 \\ 4 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 5 & 6 \end{bmatrix} \right\}
\]
**Explanation:**
- The set \( B \) contains three 2x2 matrices.
1. The first matrix is:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}
\]
This is a diagonal matrix where only the diagonal elements (1 and 2) are non-zero.
2. The second matrix is:
\[
\begin{bmatrix} 3 & 0 \\ 4 & 0 \end{bmatrix}
\]
This matrix has non-zero elements in the first column.
3. The third matrix is:
\[
\begin{bmatrix} 0 & 0 \\ 5 & 6 \end{bmatrix}
\]
This matrix has non-zero elements in the second row.
These matrices could possibly be used in linear transformations, systems of equations, or other advanced mathematical concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2F0a1f1db5-6b0c-4c01-9b03-e97ac4b2c941%2Fyma6fvj_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Set Representation:**
The set \( B \) is defined as:
\[
B = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, \begin{bmatrix} 3 & 0 \\ 4 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 5 & 6 \end{bmatrix} \right\}
\]
**Explanation:**
- The set \( B \) contains three 2x2 matrices.
1. The first matrix is:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}
\]
This is a diagonal matrix where only the diagonal elements (1 and 2) are non-zero.
2. The second matrix is:
\[
\begin{bmatrix} 3 & 0 \\ 4 & 0 \end{bmatrix}
\]
This matrix has non-zero elements in the first column.
3. The third matrix is:
\[
\begin{bmatrix} 0 & 0 \\ 5 & 6 \end{bmatrix}
\]
This matrix has non-zero elements in the second row.
These matrices could possibly be used in linear transformations, systems of equations, or other advanced mathematical concepts.
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