1. Which of the following sets of matrices form a subspace of M22? (assume a, b range across the real numbers) a b] А. 6 a [a 1 a С. 0 [o o] D. 0 0 B.
1. Which of the following sets of matrices form a subspace of M22? (assume a, b range across the real numbers) a b] А. 6 a [a 1 a С. 0 [o o] D. 0 0 B.
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
Multiple Choice
![### Linear Algebra: Subspaces of Matrices
**Question:**
Which of the following sets of matrices form a subspace of \( M_{2,2} \)? (assume \( a, b \) range across the real numbers)
**Options:**
A.
\[
\begin{bmatrix}
a & b \\
b & a
\end{bmatrix}
\]
B.
\[
\begin{bmatrix}
a & 1 \\
1 & a
\end{bmatrix}
\]
C.
\[
\begin{bmatrix}
a & 0 \\
0 & a
\end{bmatrix}
\]
D.
\[
\begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}
\]
**Explanation:**
- **Option A** presents symmetric matrices, where the off-diagonal elements are equal.
- **Option B** presents a specific type of matrix where the off-diagonal elements are always 1.
- **Option C** presents diagonal matrices with equal elements on the diagonal.
- **Option D** presents the zero matrix, which is the identity element of matrix addition.
**Discussion:**
For a set of matrices to form a subspace of \( M_{2,2} \), it must be closed under matrix addition and scalar multiplication, and it must contain the zero matrix.
- **Option A** (Symmetric Matrices): These matrices form a subspace because the set of symmetric matrices is closed under addition and scalar multiplication.
- **Option B** (Matrices with fixed off-diagonal elements of 1): This set does not form a subspace because it is not closed under scalar multiplication (scaling the matrix would not keep the off-diagonal elements as 1).
- **Option C** (Diagonal Matrices with equal diagonal elements): These matrices form a subspace as they are closed under addition and scalar multiplication.
- **Option D** (Zero Matrix): This set does form a subspace as the set containing only the zero matrix trivially satisfies all the subspace properties.
**Answer:**
A and C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f133c63-1716-41d7-8ec5-0f87428512d1%2F64f6a378-7cb6-4f45-8f2f-82d3809ed833%2Fcsd2mre_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Subspaces of Matrices
**Question:**
Which of the following sets of matrices form a subspace of \( M_{2,2} \)? (assume \( a, b \) range across the real numbers)
**Options:**
A.
\[
\begin{bmatrix}
a & b \\
b & a
\end{bmatrix}
\]
B.
\[
\begin{bmatrix}
a & 1 \\
1 & a
\end{bmatrix}
\]
C.
\[
\begin{bmatrix}
a & 0 \\
0 & a
\end{bmatrix}
\]
D.
\[
\begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}
\]
**Explanation:**
- **Option A** presents symmetric matrices, where the off-diagonal elements are equal.
- **Option B** presents a specific type of matrix where the off-diagonal elements are always 1.
- **Option C** presents diagonal matrices with equal elements on the diagonal.
- **Option D** presents the zero matrix, which is the identity element of matrix addition.
**Discussion:**
For a set of matrices to form a subspace of \( M_{2,2} \), it must be closed under matrix addition and scalar multiplication, and it must contain the zero matrix.
- **Option A** (Symmetric Matrices): These matrices form a subspace because the set of symmetric matrices is closed under addition and scalar multiplication.
- **Option B** (Matrices with fixed off-diagonal elements of 1): This set does not form a subspace because it is not closed under scalar multiplication (scaling the matrix would not keep the off-diagonal elements as 1).
- **Option C** (Diagonal Matrices with equal diagonal elements): These matrices form a subspace as they are closed under addition and scalar multiplication.
- **Option D** (Zero Matrix): This set does form a subspace as the set containing only the zero matrix trivially satisfies all the subspace properties.
**Answer:**
A and C
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
