Explain why S is not a basis for M22
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
![**Question 6: Explain why S is not a basis for M2x2.**
**Given Set:**
\[ S = \left\{ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right\} \]
**Explanation:**
To determine if the set \( S \) is a basis for \( M2x2 \), which is the space of 2x2 matrices, we need to check if \( S \) satisfies two conditions:
1. **Spans \( M2x2 \):** A basis must span the entire space of 2x2 matrices. This means any 2x2 matrix should be expressible as a linear combination of the matrices in \( S \).
2. **Linear Independence:** The matrices in \( S \) must be linearly independent, meaning no matrix in the set can be written as a linear combination of the others.
Since \( M2x2 \) is a 4-dimensional space (as it consists of 2x2 matrices, totaling 4 elements), a basis for this space must contain 4 linearly independent matrices. However, set \( S \) only contains 2 matrices.
Therefore, \( S \) cannot be a basis for \( M2x2 \) because it contains fewer matrices than the dimension of the space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4830dd89-8f3d-4b81-a0b2-1968a7068beb%2F3251f147-245c-4118-9a18-179080dcfa8d%2F9fnhoi7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 6: Explain why S is not a basis for M2x2.**
**Given Set:**
\[ S = \left\{ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right\} \]
**Explanation:**
To determine if the set \( S \) is a basis for \( M2x2 \), which is the space of 2x2 matrices, we need to check if \( S \) satisfies two conditions:
1. **Spans \( M2x2 \):** A basis must span the entire space of 2x2 matrices. This means any 2x2 matrix should be expressible as a linear combination of the matrices in \( S \).
2. **Linear Independence:** The matrices in \( S \) must be linearly independent, meaning no matrix in the set can be written as a linear combination of the others.
Since \( M2x2 \) is a 4-dimensional space (as it consists of 2x2 matrices, totaling 4 elements), a basis for this space must contain 4 linearly independent matrices. However, set \( S \) only contains 2 matrices.
Therefore, \( S \) cannot be a basis for \( M2x2 \) because it contains fewer matrices than the dimension of the space.
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 2 steps with 1 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,
