9. Show that the matrices ( ) ( ) (19). generate M₂×2(F). and (689)
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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![**Problem 9: Matrix Generation**
Show that the matrices
\[
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
,
\begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}
,
\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}
, \text{ and }
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
\]
generate \( M_{2 \times 2}(F) \).
---
Explanation:
The problem asks to show that the given 2x2 matrices serve as generators for all 2x2 matrices over the field \( F \). This involves demonstrating that any 2x2 matrix
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
can be expressed as a linear combination of the given matrices.
Each matrix has a single '1' in different positions and zeros elsewhere, representing the standard basis for the space of 2x2 matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ada9a20-cfc4-4295-96e5-fd4aa8bf1cec%2Fbced4f92-6659-4fa5-964f-e97716712941%2Fqbvrkt9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 9: Matrix Generation**
Show that the matrices
\[
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
,
\begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}
,
\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}
, \text{ and }
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
\]
generate \( M_{2 \times 2}(F) \).
---
Explanation:
The problem asks to show that the given 2x2 matrices serve as generators for all 2x2 matrices over the field \( F \). This involves demonstrating that any 2x2 matrix
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
can be expressed as a linear combination of the given matrices.
Each matrix has a single '1' in different positions and zeros elsewhere, representing the standard basis for the space of 2x2 matrices.
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