5.15. Let A = = | 11B-2 1-1 1 = C = set of matrices {A, B, C, D} form a basis for the vector space M₂x2? and D = | -2 2: Do the
5.15. Let A = = | 11B-2 1-1 1 = C = set of matrices {A, B, C, D} form a basis for the vector space M₂x2? and D = | -2 2: Do the
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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Can you do this the matrix way only I uploaded the question and answer
![### Linear Algebra - Problem 5.15
#### Problem Statement:
Let the matrices be defined as follows:
\[
A = \begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix}, \quad
B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}, \quad
C = \begin{bmatrix} -1 & 2 \\ 1 & 1 \end{bmatrix}, \quad
D = \begin{bmatrix} 1 & 1 \\ -2 & 1 \end{bmatrix}
\]
Do the set of matrices \(\{A, B, C, D\}\) form a basis for the vector space \( M_{2 \times 2} \)?
#### Explanation:
This question focuses on determining whether the given set of matrices \(\{A, B, C, D\}\) forms a basis for the vector space of all \(2 \times 2\) matrices, denoted \( M_{2 \times 2} \).
To do this, we need to check:
1. **Linear Independence**: The set \(\{A, B, C, D\}\) must be linearly independent.
2. **Spanning the Space**: The set must span \( M_{2 \times 2} \).
### Steps:
1. **Arrange the matrices into a matrix equation**.
2. **Check Linear Independence** by determining if there is a non-trivial solution to the equation \(c_1A + c_2B + c_3C + c_4D = 0\).
3. **Span the Space**: Verify the rank to ensure it equals 4, which is the dimension of \(M_{2 \times 2}\).
By understanding these matrices and their linear properties, one can confirm whether they form a suitable basis for the space \( M_{2 \times 2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6f07dba-75ce-4887-9617-f6bb8f9dcf1f%2F30a08e41-88e8-4f69-951a-93f9cc8556ad%2Fembuhc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra - Problem 5.15
#### Problem Statement:
Let the matrices be defined as follows:
\[
A = \begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix}, \quad
B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}, \quad
C = \begin{bmatrix} -1 & 2 \\ 1 & 1 \end{bmatrix}, \quad
D = \begin{bmatrix} 1 & 1 \\ -2 & 1 \end{bmatrix}
\]
Do the set of matrices \(\{A, B, C, D\}\) form a basis for the vector space \( M_{2 \times 2} \)?
#### Explanation:
This question focuses on determining whether the given set of matrices \(\{A, B, C, D\}\) forms a basis for the vector space of all \(2 \times 2\) matrices, denoted \( M_{2 \times 2} \).
To do this, we need to check:
1. **Linear Independence**: The set \(\{A, B, C, D\}\) must be linearly independent.
2. **Spanning the Space**: The set must span \( M_{2 \times 2} \).
### Steps:
1. **Arrange the matrices into a matrix equation**.
2. **Check Linear Independence** by determining if there is a non-trivial solution to the equation \(c_1A + c_2B + c_3C + c_4D = 0\).
3. **Span the Space**: Verify the rank to ensure it equals 4, which is the dimension of \(M_{2 \times 2}\).
By understanding these matrices and their linear properties, one can confirm whether they form a suitable basis for the space \( M_{2 \times 2} \).
Transcribed Image Text:**5.15 Yes. Show that they are linearly independent. Since \( M_{2 \times 2} \) has dimension 4, hence four linearly independent vectors form a basis.**
Explanation:
The text addresses the concept of linear independence in the context of vector spaces. It mentions a particular solution (5.15) and states that to show four vectors are linearly independent, one must demonstrate that they form a basis. Specifically, for the vector space \( M_{2 \times 2} \) (the space of all 2x2 matrices), which has a dimension of 4, four linearly independent matrices would form a basis for this space.
Graph or Diagram Explanation:
There are no graphs or diagrams in this image, only textual content that pertains to the theoretical aspect of linear independence and basis formation in the context of matrix spaces.
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