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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
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 \( M_{2 \times 2} \).
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 \( M_{2 \times 2} \).
**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.
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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