1 2 0 0 2 0 0 2 0 0 plain why span M2(R) 0 4 3 4
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
![### Explanation of Matrix Span
**Problem Statement:**
Explain why the span of the matrices
\[
\left\{
\begin{bmatrix}
2 & 0 \\
0 & 4
\end{bmatrix},
\begin{bmatrix}
0 & 2 \\
4 & 0
\end{bmatrix},
\begin{bmatrix}
1 & 2 \\
0 & 0
\end{bmatrix},
\begin{bmatrix}
0 & 0 \\
3 & 4
\end{bmatrix}
\right\}
= M_2(\mathbb{R})
\]
**Explanation:**
To prove that the set of given matrices spans \( M_2(\mathbb{R}) \), the space of all \( 2 \times 2 \) matrices with real number entries, we need to show that any \( 2 \times 2 \) matrix can be expressed as a linear combination of these four matrices.
### Process:
1. **Identify the Basis Requirement:**
- A general \( 2 \times 2 \) matrix is of the form:
\[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]
- We must find real numbers \( \alpha, \beta, \gamma, \delta \) such that:
\[
\alpha
\begin{bmatrix}
2 & 0 \\
0 & 4
\end{bmatrix}
+ \beta
\begin{bmatrix}
0 & 2 \\
4 & 0
\end{bmatrix}
+ \gamma
\begin{bmatrix}
1 & 2 \\
0 & 0
\end{bmatrix}
+ \delta
\begin{bmatrix}
0 & 0 \\
3 & 4
\end{bmatrix}
=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]
2. **Solve for Coefficients:**
- Equating the entries of matrices, we get:
- \( 2\alpha + \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdfa965ce-96ce-41c8-84ac-4edd317a194a%2F0e847521-c4e8-453a-8f5b-072e51bf74a6%2Fbqbkpes_processed.png&w=3840&q=75)
Transcribed Image Text:### Explanation of Matrix Span
**Problem Statement:**
Explain why the span of the matrices
\[
\left\{
\begin{bmatrix}
2 & 0 \\
0 & 4
\end{bmatrix},
\begin{bmatrix}
0 & 2 \\
4 & 0
\end{bmatrix},
\begin{bmatrix}
1 & 2 \\
0 & 0
\end{bmatrix},
\begin{bmatrix}
0 & 0 \\
3 & 4
\end{bmatrix}
\right\}
= M_2(\mathbb{R})
\]
**Explanation:**
To prove that the set of given matrices spans \( M_2(\mathbb{R}) \), the space of all \( 2 \times 2 \) matrices with real number entries, we need to show that any \( 2 \times 2 \) matrix can be expressed as a linear combination of these four matrices.
### Process:
1. **Identify the Basis Requirement:**
- A general \( 2 \times 2 \) matrix is of the form:
\[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]
- We must find real numbers \( \alpha, \beta, \gamma, \delta \) such that:
\[
\alpha
\begin{bmatrix}
2 & 0 \\
0 & 4
\end{bmatrix}
+ \beta
\begin{bmatrix}
0 & 2 \\
4 & 0
\end{bmatrix}
+ \gamma
\begin{bmatrix}
1 & 2 \\
0 & 0
\end{bmatrix}
+ \delta
\begin{bmatrix}
0 & 0 \\
3 & 4
\end{bmatrix}
=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]
2. **Solve for Coefficients:**
- Equating the entries of matrices, we get:
- \( 2\alpha + \
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