Determine whether the set of vectors in M2.2 is linearly independent or linearly dependent. 2 -8 A = C = 22 23 O linearly independent O linearly dependent
Determine whether the set of vectors in M2.2 is linearly independent or linearly dependent. 2 -8 A = C = 22 23 O linearly independent O linearly dependent
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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Question
![**Problem Statement:**
Determine whether the set of vectors in \( M_{2,2} \) is linearly independent or linearly dependent.
**Given Matrices:**
\[
A = \begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix}, \quad
B = \begin{bmatrix} 5 & 3 \\ -3 & 3 \end{bmatrix}, \quad
C = \begin{bmatrix} 2 & -8 \\ 22 & 23 \end{bmatrix}
\]
**Options:**
- ⬤ Linearly independent
- ⬤ Linearly dependent
**Explanation:**
In this problem, you are asked to determine the linear dependence or independence of a set of given matrices. Each matrix is a \(2 \times 2\) matrix, represented by \(A\), \(B\), and \(C\).
- **Matrix A** is defined as having elements \([2, -1]\) in the first row, and \([4, 5]\) in the second row.
- **Matrix B** contains elements \([5, 3]\) and \([-3, 3]\).
- **Matrix C** includes \([2, -8]\) and \([22, 23]\).
To determine if the set is linearly independent or dependent, you generally check if any matrix can be written as a linear combination of the others. If this is possible, the matrices are linearly dependent; if not, they are linearly independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a70e2d2-d641-435a-b212-d4266cff05fc%2F01e53393-607e-4238-920a-a96af31aa7e7%2F9zhr52l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the set of vectors in \( M_{2,2} \) is linearly independent or linearly dependent.
**Given Matrices:**
\[
A = \begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix}, \quad
B = \begin{bmatrix} 5 & 3 \\ -3 & 3 \end{bmatrix}, \quad
C = \begin{bmatrix} 2 & -8 \\ 22 & 23 \end{bmatrix}
\]
**Options:**
- ⬤ Linearly independent
- ⬤ Linearly dependent
**Explanation:**
In this problem, you are asked to determine the linear dependence or independence of a set of given matrices. Each matrix is a \(2 \times 2\) matrix, represented by \(A\), \(B\), and \(C\).
- **Matrix A** is defined as having elements \([2, -1]\) in the first row, and \([4, 5]\) in the second row.
- **Matrix B** contains elements \([5, 3]\) and \([-3, 3]\).
- **Matrix C** includes \([2, -8]\) and \([22, 23]\).
To determine if the set is linearly independent or dependent, you generally check if any matrix can be written as a linear combination of the others. If this is possible, the matrices are linearly dependent; if not, they are linearly independent.
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