Is this a subspace of M
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 subspace of M2×2?
![The image displays a mathematical expression defining a set \( V \):
\[
V = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \bigg| a + b = 0, \, c \in \mathbb{R} \right\}
\]
### Explanation
- **Set Notation**: The curly braces \(\{\}\) are used to define the set \( V \).
- **Matrix Structure**: The elements of the set are matrices of size 2x2 with the form:
\[
\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}
\]
Here, \( a \), \( b \), and \( c \) are real numbers, denoted as \( a, b, c \in \mathbb{R} \).
- **Condition**: A condition given is \( a + b = 0 \). This implies that for the matrix to be included in the set \( V \), the sum of elements \( a \) and \( b \), which are in the first row, must be zero.
- **Variable \( c \)**: The element \( c \) in the second row can be any real number, indicated by \( c \in \mathbb{R} \).
This type of mathematical expression is typically used in linear algebra to define specific subspaces or sets of matrices that satisfy given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2Fe8b4c267-e5c5-4e38-8276-30bf5b8b908d%2F44p1gm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical expression defining a set \( V \):
\[
V = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \bigg| a + b = 0, \, c \in \mathbb{R} \right\}
\]
### Explanation
- **Set Notation**: The curly braces \(\{\}\) are used to define the set \( V \).
- **Matrix Structure**: The elements of the set are matrices of size 2x2 with the form:
\[
\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}
\]
Here, \( a \), \( b \), and \( c \) are real numbers, denoted as \( a, b, c \in \mathbb{R} \).
- **Condition**: A condition given is \( a + b = 0 \). This implies that for the matrix to be included in the set \( V \), the sum of elements \( a \) and \( b \), which are in the first row, must be zero.
- **Variable \( c \)**: The element \( c \) in the second row can be any real number, indicated by \( c \in \mathbb{R} \).
This type of mathematical expression is typically used in linear algebra to define specific subspaces or sets of matrices that satisfy given conditions.
Expert Solution
Step 1
Introduction:
A vector space that is totally contained within another vector space is known as a subspace. Both are required to completely define a subspace because it is defined relative to its contained space; for instance, is a subspace of as well as etc.
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